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stringclasses 18
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1
|
1918-1921
|
(ii)
As a special case of (4) above, if f is the constant function f (x) = λ, then the
function g
λ defined by
( )
( )
gx
g x
λ
λ
=
is also continuous wherever g(x) ≠ 0 In
particular, the continuity of g implies continuity of 1
g The above theorem can be exploited to generate many continuous functions They
also aid in deciding if certain functions are continuous or not
|
1
|
1919-1922
|
In
particular, the continuity of g implies continuity of 1
g The above theorem can be exploited to generate many continuous functions They
also aid in deciding if certain functions are continuous or not The following examples
illustrate this:
Example 16 Prove that every rational function is continuous
|
1
|
1920-1923
|
The above theorem can be exploited to generate many continuous functions They
also aid in deciding if certain functions are continuous or not The following examples
illustrate this:
Example 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by
( )
( )
,
( )
0
( )
p x
f x
q x
=q x
≠
where p and q are polynomial functions
|
1
|
1921-1924
|
They
also aid in deciding if certain functions are continuous or not The following examples
illustrate this:
Example 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by
( )
( )
,
( )
0
( )
p x
f x
q x
=q x
≠
where p and q are polynomial functions The domain of f is all real numbers except
points at which q is zero
|
1
|
1922-1925
|
The following examples
illustrate this:
Example 16 Prove that every rational function is continuous Solution Recall that every rational function f is given by
( )
( )
,
( )
0
( )
p x
f x
q x
=q x
≠
where p and q are polynomial functions The domain of f is all real numbers except
points at which q is zero Since polynomial functions are continuous (Example 14), f is
continuous by (4) of Theorem 1
|
1
|
1923-1926
|
Solution Recall that every rational function f is given by
( )
( )
,
( )
0
( )
p x
f x
q x
=q x
≠
where p and q are polynomial functions The domain of f is all real numbers except
points at which q is zero Since polynomial functions are continuous (Example 14), f is
continuous by (4) of Theorem 1 Example 17 Discuss the continuity of sine function
|
1
|
1924-1927
|
The domain of f is all real numbers except
points at which q is zero Since polynomial functions are continuous (Example 14), f is
continuous by (4) of Theorem 1 Example 17 Discuss the continuity of sine function Solution To see this we use the following facts
0
lim sin
0
x
x
→
=
We have not proved it, but is intuitively clear from the graph of sin x near 0
|
1
|
1925-1928
|
Since polynomial functions are continuous (Example 14), f is
continuous by (4) of Theorem 1 Example 17 Discuss the continuity of sine function Solution To see this we use the following facts
0
lim sin
0
x
x
→
=
We have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number
|
1
|
1926-1929
|
Example 17 Discuss the continuity of sine function Solution To see this we use the following facts
0
lim sin
0
x
x
→
=
We have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number Let c be a real
number
|
1
|
1927-1930
|
Solution To see this we use the following facts
0
lim sin
0
x
x
→
=
We have not proved it, but is intuitively clear from the graph of sin x near 0 Now, observe that f (x) = sin x is defined for every real number Let c be a real
number Put x = c + h
|
1
|
1928-1931
|
Now, observe that f (x) = sin x is defined for every real number Let c be a real
number Put x = c + h If x → c we know that h → 0
|
1
|
1929-1932
|
Let c be a real
number Put x = c + h If x → c we know that h → 0 Therefore
lim
( )
x
c f x
→
= lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→ f (x) = f(c) and hence f is a continuous function
|
1
|
1930-1933
|
Put x = c + h If x → c we know that h → 0 Therefore
lim
( )
x
c f x
→
= lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→ f (x) = f(c) and hence f is a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function
|
1
|
1931-1934
|
If x → c we know that h → 0 Therefore
lim
( )
x
c f x
→
= lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→ f (x) = f(c) and hence f is a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function
|
1
|
1932-1935
|
Therefore
lim
( )
x
c f x
→
= lim sin
x
c
x
→
=
0
lim sin(
)
h
c
h
→
+
=
0
lim [sin cos
cos sin ]
h
c
h
c
h
→
+
=
0
0
lim [sin cos ]
lim [cos sin ]
h
h
c
h
c
h
→
→
+
= sin c + 0 = sin c = f (c)
Thus lim
x
c
→ f (x) = f(c) and hence f is a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin
cos
x
x
|
1
|
1933-1936
|
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
115
Remark A similar proof may be given for the continuity of cosine function Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin
cos
x
x This is defined for all real numbers such
that cos x ≠ 0, i
|
1
|
1934-1937
|
Example 18 Prove that the function defined by f(x) = tan x is a continuous function Solution The function f (x) = tan x = sin
cos
x
x This is defined for all real numbers such
that cos x ≠ 0, i e
|
1
|
1935-1938
|
Solution The function f (x) = tan x = sin
cos
x
x This is defined for all real numbers such
that cos x ≠ 0, i e , x ≠ (2n +1) 2
π
|
1
|
1936-1939
|
This is defined for all real numbers such
that cos x ≠ 0, i e , x ≠ (2n +1) 2
π We have just proved that both sine and cosine
functions are continuous
|
1
|
1937-1940
|
e , x ≠ (2n +1) 2
π We have just proved that both sine and cosine
functions are continuous Thus tan x being a quotient of two continuous functions is
continuous wherever it is defined
|
1
|
1938-1941
|
, x ≠ (2n +1) 2
π We have just proved that both sine and cosine
functions are continuous Thus tan x being a quotient of two continuous functions is
continuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to
composition of functions
|
1
|
1939-1942
|
We have just proved that both sine and cosine
functions are continuous Thus tan x being a quotient of two continuous functions is
continuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to
composition of functions Recall that if f and g are two real functions, then
(f o g) (x) = f(g (x))
is defined whenever the range of g is a subset of domain of f
|
1
|
1940-1943
|
Thus tan x being a quotient of two continuous functions is
continuous wherever it is defined An interesting fact is the behaviour of continuous functions with respect to
composition of functions Recall that if f and g are two real functions, then
(f o g) (x) = f(g (x))
is defined whenever the range of g is a subset of domain of f The following theorem
(stated without proof) captures the continuity of composite functions
|
1
|
1941-1944
|
An interesting fact is the behaviour of continuous functions with respect to
composition of functions Recall that if f and g are two real functions, then
(f o g) (x) = f(g (x))
is defined whenever the range of g is a subset of domain of f The following theorem
(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c
|
1
|
1942-1945
|
Recall that if f and g are two real functions, then
(f o g) (x) = f(g (x))
is defined whenever the range of g is a subset of domain of f The following theorem
(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c
|
1
|
1943-1946
|
The following theorem
(stated without proof) captures the continuity of composite functions Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem
|
1
|
1944-1947
|
Theorem 2 Suppose f and g are real valued functions such that (f o g) is defined at c If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function
|
1
|
1945-1948
|
If g is continuous at c and if f is continuous at g(c), then (f o g) is continuous at c The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number
|
1
|
1946-1949
|
The following examples illustrate this theorem Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number The function
f may be thought of as a composition g o h of the two functions g and h, where
g (x) = sin x and h(x) = x2
|
1
|
1947-1950
|
Example 19 Show that the function defined by f(x) = sin (x2) is a continuous function Solution Observe that the function is defined for every real number The function
f may be thought of as a composition g o h of the two functions g and h, where
g (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,
it can be deduced that f is a continuous function
|
1
|
1948-1951
|
Solution Observe that the function is defined for every real number The function
f may be thought of as a composition g o h of the two functions g and h, where
g (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,
it can be deduced that f is a continuous function Example 20 Show that the function f defined by
f (x) = |1 – x + | x||,
where x is any real number, is a continuous function
|
1
|
1949-1952
|
The function
f may be thought of as a composition g o h of the two functions g and h, where
g (x) = sin x and h(x) = x2 Since both g and h are continuous functions, by Theorem 2,
it can be deduced that f is a continuous function Example 20 Show that the function f defined by
f (x) = |1 – x + | x||,
where x is any real number, is a continuous function Solution Define g by g(x) = 1 – x + |x| and h by h(x) = |x| for all real x
|
1
|
1950-1953
|
Since both g and h are continuous functions, by Theorem 2,
it can be deduced that f is a continuous function Example 20 Show that the function f defined by
f (x) = |1 – x + | x||,
where x is any real number, is a continuous function Solution Define g by g(x) = 1 – x + |x| and h by h(x) = |x| for all real x Then
(h o g) (x) = h (g (x))
= h (1– x + |x |)
= |1– x + | x|| = f (x)
In Example 7, we have seen that h is a continuous function
|
1
|
1951-1954
|
Example 20 Show that the function f defined by
f (x) = |1 – x + | x||,
where x is any real number, is a continuous function Solution Define g by g(x) = 1 – x + |x| and h by h(x) = |x| for all real x Then
(h o g) (x) = h (g (x))
= h (1– x + |x |)
= |1– x + | x|| = f (x)
In Example 7, we have seen that h is a continuous function Hence g being a sum
of a polynomial function and the modulus function is continuous
|
1
|
1952-1955
|
Solution Define g by g(x) = 1 – x + |x| and h by h(x) = |x| for all real x Then
(h o g) (x) = h (g (x))
= h (1– x + |x |)
= |1– x + | x|| = f (x)
In Example 7, we have seen that h is a continuous function Hence g being a sum
of a polynomial function and the modulus function is continuous But then f being a
composite of two continuous functions is continuous
|
1
|
1953-1956
|
Then
(h o g) (x) = h (g (x))
= h (1– x + |x |)
= |1– x + | x|| = f (x)
In Example 7, we have seen that h is a continuous function Hence g being a sum
of a polynomial function and the modulus function is continuous But then f being a
composite of two continuous functions is continuous Rationalised 2023-24
MATHEMATICS
116
EXERCISE 5
|
1
|
1954-1957
|
Hence g being a sum
of a polynomial function and the modulus function is continuous But then f being a
composite of two continuous functions is continuous Rationalised 2023-24
MATHEMATICS
116
EXERCISE 5 1
1
|
1
|
1955-1958
|
But then f being a
composite of two continuous functions is continuous Rationalised 2023-24
MATHEMATICS
116
EXERCISE 5 1
1 Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5
|
1
|
1956-1959
|
Rationalised 2023-24
MATHEMATICS
116
EXERCISE 5 1
1 Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5 2
|
1
|
1957-1960
|
1
1 Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5 2 Examine the continuity of the function f(x) = 2x2 – 1 at x = 3
|
1
|
1958-1961
|
Prove that the function f(x) = 5x – 3 is continuous at x = 0, at x = – 3 and at x = 5 2 Examine the continuity of the function f(x) = 2x2 – 1 at x = 3 3
|
1
|
1959-1962
|
2 Examine the continuity of the function f(x) = 2x2 – 1 at x = 3 3 Examine the following functions for continuity
|
1
|
1960-1963
|
Examine the continuity of the function f(x) = 2x2 – 1 at x = 3 3 Examine the following functions for continuity (a)
f(x) = x – 5
(b)
f(x) =
1
x −5
, x ≠ 5
(c)
f(x) =
2
25
5
x
x
+−
, x ≠ –5
(d)
f(x) = |x – 5 |
4
|
1
|
1961-1964
|
3 Examine the following functions for continuity (a)
f(x) = x – 5
(b)
f(x) =
1
x −5
, x ≠ 5
(c)
f(x) =
2
25
5
x
x
+−
, x ≠ –5
(d)
f(x) = |x – 5 |
4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive
integer
|
1
|
1962-1965
|
Examine the following functions for continuity (a)
f(x) = x – 5
(b)
f(x) =
1
x −5
, x ≠ 5
(c)
f(x) =
2
25
5
x
x
+−
, x ≠ –5
(d)
f(x) = |x – 5 |
4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive
integer 5
|
1
|
1963-1966
|
(a)
f(x) = x – 5
(b)
f(x) =
1
x −5
, x ≠ 5
(c)
f(x) =
2
25
5
x
x
+−
, x ≠ –5
(d)
f(x) = |x – 5 |
4 Prove that the function f(x) = xn is continuous at x = n, where n is a positive
integer 5 Is the function f defined by
, if
1
( )
5, if
>1
x
x
f x
x
≤
=
continuous at x = 0
|
1
|
1964-1967
|
Prove that the function f(x) = xn is continuous at x = n, where n is a positive
integer 5 Is the function f defined by
, if
1
( )
5, if
>1
x
x
f x
x
≤
=
continuous at x = 0 At x = 1
|
1
|
1965-1968
|
5 Is the function f defined by
, if
1
( )
5, if
>1
x
x
f x
x
≤
=
continuous at x = 0 At x = 1 At x = 2
|
1
|
1966-1969
|
Is the function f defined by
, if
1
( )
5, if
>1
x
x
f x
x
≤
=
continuous at x = 0 At x = 1 At x = 2 Find all points of discontinuity of f, where f is defined by
6
|
1
|
1967-1970
|
At x = 1 At x = 2 Find all points of discontinuity of f, where f is defined by
6 2
3, if
2
( )
2
3, if
> 2
x
x
f x
x
x
+
≤
=
−
7
|
1
|
1968-1971
|
At x = 2 Find all points of discontinuity of f, where f is defined by
6 2
3, if
2
( )
2
3, if
> 2
x
x
f x
x
x
+
≤
=
−
7 |
| 3, if
3
( )
2 , if
3
< 3
6
2, if
3
x
x
f x
x
x
x
x
+
≤ −
=
−
−
<
+
≥
8
|
1
|
1969-1972
|
Find all points of discontinuity of f, where f is defined by
6 2
3, if
2
( )
2
3, if
> 2
x
x
f x
x
x
+
≤
=
−
7 |
| 3, if
3
( )
2 , if
3
< 3
6
2, if
3
x
x
f x
x
x
x
x
+
≤ −
=
−
−
<
+
≥
8 |
|, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
9
|
1
|
1970-1973
|
2
3, if
2
( )
2
3, if
> 2
x
x
f x
x
x
+
≤
=
−
7 |
| 3, if
3
( )
2 , if
3
< 3
6
2, if
3
x
x
f x
x
x
x
x
+
≤ −
=
−
−
<
+
≥
8 |
|, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
9 , if
0
|
|
( )
1,
if
0
x
x
x
f x
x
<
=
−
≥
10
|
1
|
1971-1974
|
|
| 3, if
3
( )
2 , if
3
< 3
6
2, if
3
x
x
f x
x
x
x
x
+
≤ −
=
−
−
<
+
≥
8 |
|, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
9 , if
0
|
|
( )
1,
if
0
x
x
x
f x
x
<
=
−
≥
10 2
1, if
1
( )
1, if
1
x
x
f x
x
x
+
≥
=
+
<
11
|
1
|
1972-1975
|
|
|, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
9 , if
0
|
|
( )
1,
if
0
x
x
x
f x
x
<
=
−
≥
10 2
1, if
1
( )
1, if
1
x
x
f x
x
x
+
≥
=
+
<
11 3
2
3, if
2
( )
1,
if
2
x
x
f x
x
x
−
≤
=
+
>
12
|
1
|
1973-1976
|
, if
0
|
|
( )
1,
if
0
x
x
x
f x
x
<
=
−
≥
10 2
1, if
1
( )
1, if
1
x
x
f x
x
x
+
≥
=
+
<
11 3
2
3, if
2
( )
1,
if
2
x
x
f x
x
x
−
≤
=
+
>
12 10
2
1, if
1
( )
,
if
1
x
x
f x
x
x
−
≤
=
>
13
|
1
|
1974-1977
|
2
1, if
1
( )
1, if
1
x
x
f x
x
x
+
≥
=
+
<
11 3
2
3, if
2
( )
1,
if
2
x
x
f x
x
x
−
≤
=
+
>
12 10
2
1, if
1
( )
,
if
1
x
x
f x
x
x
−
≤
=
>
13 Is the function defined by
5, if
1
( )
5, if
1
x
x
f x
x
x
+
≤
= −
>
a continuous function
|
1
|
1975-1978
|
3
2
3, if
2
( )
1,
if
2
x
x
f x
x
x
−
≤
=
+
>
12 10
2
1, if
1
( )
,
if
1
x
x
f x
x
x
−
≤
=
>
13 Is the function defined by
5, if
1
( )
5, if
1
x
x
f x
x
x
+
≤
= −
>
a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
117
Discuss the continuity of the function f, where f is defined by
14
|
1
|
1976-1979
|
10
2
1, if
1
( )
,
if
1
x
x
f x
x
x
−
≤
=
>
13 Is the function defined by
5, if
1
( )
5, if
1
x
x
f x
x
x
+
≤
= −
>
a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
117
Discuss the continuity of the function f, where f is defined by
14 3, if 0
1
( )
4, if 1
3
5, if 3
10
x
f x
x
x
≤
≤
=
<
<
≤
≤
15
|
1
|
1977-1980
|
Is the function defined by
5, if
1
( )
5, if
1
x
x
f x
x
x
+
≤
= −
>
a continuous function Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
117
Discuss the continuity of the function f, where f is defined by
14 3, if 0
1
( )
4, if 1
3
5, if 3
10
x
f x
x
x
≤
≤
=
<
<
≤
≤
15 2 , if
0
( )
0,
if 0
1
4 , if
>1
x
x
f x
x
x
x
<
=
≤
≤
16
|
1
|
1978-1981
|
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
117
Discuss the continuity of the function f, where f is defined by
14 3, if 0
1
( )
4, if 1
3
5, if 3
10
x
f x
x
x
≤
≤
=
<
<
≤
≤
15 2 , if
0
( )
0,
if 0
1
4 , if
>1
x
x
f x
x
x
x
<
=
≤
≤
16 2, if
1
( )
2 , if
1
1
2,
if
1
x
f x
x
x
x
−
≤ −
=
− <
≤
>
17
|
1
|
1979-1982
|
3, if 0
1
( )
4, if 1
3
5, if 3
10
x
f x
x
x
≤
≤
=
<
<
≤
≤
15 2 , if
0
( )
0,
if 0
1
4 , if
>1
x
x
f x
x
x
x
<
=
≤
≤
16 2, if
1
( )
2 , if
1
1
2,
if
1
x
f x
x
x
x
−
≤ −
=
− <
≤
>
17 Find the relationship between a and b so that the function f defined by
1, if
3
( )
3, if
3
ax
x
f x
bx
x
+
≤
=
+
>
is continuous at x = 3
|
1
|
1980-1983
|
2 , if
0
( )
0,
if 0
1
4 , if
>1
x
x
f x
x
x
x
<
=
≤
≤
16 2, if
1
( )
2 , if
1
1
2,
if
1
x
f x
x
x
x
−
≤ −
=
− <
≤
>
17 Find the relationship between a and b so that the function f defined by
1, if
3
( )
3, if
3
ax
x
f x
bx
x
+
≤
=
+
>
is continuous at x = 3 18
|
1
|
1981-1984
|
2, if
1
( )
2 , if
1
1
2,
if
1
x
f x
x
x
x
−
≤ −
=
− <
≤
>
17 Find the relationship between a and b so that the function f defined by
1, if
3
( )
3, if
3
ax
x
f x
bx
x
+
≤
=
+
>
is continuous at x = 3 18 For what value of λ is the function defined by
(2
2 ), if
0
( )
4
1,
if
0
x
x
x
f x
x
x
λ
−
≤
=
+
>
continuous at x = 0
|
1
|
1982-1985
|
Find the relationship between a and b so that the function f defined by
1, if
3
( )
3, if
3
ax
x
f x
bx
x
+
≤
=
+
>
is continuous at x = 3 18 For what value of λ is the function defined by
(2
2 ), if
0
( )
4
1,
if
0
x
x
x
f x
x
x
λ
−
≤
=
+
>
continuous at x = 0 What about continuity at x = 1
|
1
|
1983-1986
|
18 For what value of λ is the function defined by
(2
2 ), if
0
( )
4
1,
if
0
x
x
x
f x
x
x
λ
−
≤
=
+
>
continuous at x = 0 What about continuity at x = 1 19
|
1
|
1984-1987
|
For what value of λ is the function defined by
(2
2 ), if
0
( )
4
1,
if
0
x
x
x
f x
x
x
λ
−
≤
=
+
>
continuous at x = 0 What about continuity at x = 1 19 Show that the function defined by g (x) = x – [x] is discontinuous at all integral
points
|
1
|
1985-1988
|
What about continuity at x = 1 19 Show that the function defined by g (x) = x – [x] is discontinuous at all integral
points Here [x] denotes the greatest integer less than or equal to x
|
1
|
1986-1989
|
19 Show that the function defined by g (x) = x – [x] is discontinuous at all integral
points Here [x] denotes the greatest integer less than or equal to x 20
|
1
|
1987-1990
|
Show that the function defined by g (x) = x – [x] is discontinuous at all integral
points Here [x] denotes the greatest integer less than or equal to x 20 Is the function defined by f(x) = x2 – sin x + 5 continuous at x = π
|
1
|
1988-1991
|
Here [x] denotes the greatest integer less than or equal to x 20 Is the function defined by f(x) = x2 – sin x + 5 continuous at x = π 21
|
1
|
1989-1992
|
20 Is the function defined by f(x) = x2 – sin x + 5 continuous at x = π 21 Discuss the continuity of the following functions:
(a)
f(x) = sin x + cos x
(b)
f(x) = sin x – cos x
(c)
f(x) = sin x
|
1
|
1990-1993
|
Is the function defined by f(x) = x2 – sin x + 5 continuous at x = π 21 Discuss the continuity of the following functions:
(a)
f(x) = sin x + cos x
(b)
f(x) = sin x – cos x
(c)
f(x) = sin x cos x
22
|
1
|
1991-1994
|
21 Discuss the continuity of the following functions:
(a)
f(x) = sin x + cos x
(b)
f(x) = sin x – cos x
(c)
f(x) = sin x cos x
22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions
|
1
|
1992-1995
|
Discuss the continuity of the following functions:
(a)
f(x) = sin x + cos x
(b)
f(x) = sin x – cos x
(c)
f(x) = sin x cos x
22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23
|
1
|
1993-1996
|
cos x
22 Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23 Find all points of discontinuity of f, where
sin
, if
0
( )
1,
if
0
x
x
f x
xx
x
<
=
+
≥
24
|
1
|
1994-1997
|
Discuss the continuity of the cosine, cosecant, secant and cotangent functions 23 Find all points of discontinuity of f, where
sin
, if
0
( )
1,
if
0
x
x
f x
xx
x
<
=
+
≥
24 Determine if f defined by
2
sin1
, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
is a continuous function
|
1
|
1995-1998
|
23 Find all points of discontinuity of f, where
sin
, if
0
( )
1,
if
0
x
x
f x
xx
x
<
=
+
≥
24 Determine if f defined by
2
sin1
, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
is a continuous function Rationalised 2023-24
MATHEMATICS
118
25
|
1
|
1996-1999
|
Find all points of discontinuity of f, where
sin
, if
0
( )
1,
if
0
x
x
f x
xx
x
<
=
+
≥
24 Determine if f defined by
2
sin1
, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
is a continuous function Rationalised 2023-24
MATHEMATICS
118
25 Examine the continuity of f, where f is defined by
sin
cos , if
0
( )
1,
if
0
x
x
x
f x
x
−
≠
= −
=
Find the values of k so that the function f is continuous at the indicated point in Exercises
26 to 29
|
1
|
1997-2000
|
Determine if f defined by
2
sin1
, if
0
( )
0,
if
0
x
x
f x
x
x
≠
=
=
is a continuous function Rationalised 2023-24
MATHEMATICS
118
25 Examine the continuity of f, where f is defined by
sin
cos , if
0
( )
1,
if
0
x
x
x
f x
x
−
≠
= −
=
Find the values of k so that the function f is continuous at the indicated point in Exercises
26 to 29 26
|
1
|
1998-2001
|
Rationalised 2023-24
MATHEMATICS
118
25 Examine the continuity of f, where f is defined by
sin
cos , if
0
( )
1,
if
0
x
x
x
f x
x
−
≠
= −
=
Find the values of k so that the function f is continuous at the indicated point in Exercises
26 to 29 26 cos , if
2
2
( )
3,
if
2
k
x
x
x
f x
x
π
≠
= π −
π
=
at x = 2
π
27
|
1
|
1999-2002
|
Examine the continuity of f, where f is defined by
sin
cos , if
0
( )
1,
if
0
x
x
x
f x
x
−
≠
= −
=
Find the values of k so that the function f is continuous at the indicated point in Exercises
26 to 29 26 cos , if
2
2
( )
3,
if
2
k
x
x
x
f x
x
π
≠
= π −
π
=
at x = 2
π
27 2, if
2
( )
3,
if
2
kx
x
f x
x
≤
=
>
at x = 2
28
|
1
|
2000-2003
|
26 cos , if
2
2
( )
3,
if
2
k
x
x
x
f x
x
π
≠
= π −
π
=
at x = 2
π
27 2, if
2
( )
3,
if
2
kx
x
f x
x
≤
=
>
at x = 2
28 1, if
( )
cos ,
if
kx
x
f x
x
x
+
≤ π
=
> π
at x = π
29
|
1
|
2001-2004
|
cos , if
2
2
( )
3,
if
2
k
x
x
x
f x
x
π
≠
= π −
π
=
at x = 2
π
27 2, if
2
( )
3,
if
2
kx
x
f x
x
≤
=
>
at x = 2
28 1, if
( )
cos ,
if
kx
x
f x
x
x
+
≤ π
=
> π
at x = π
29 1, if
5
( )
3
5, if
5
kx
x
f x
x
x
+
≤
=
−
>
at x = 5
30
|
1
|
2002-2005
|
2, if
2
( )
3,
if
2
kx
x
f x
x
≤
=
>
at x = 2
28 1, if
( )
cos ,
if
kx
x
f x
x
x
+
≤ π
=
> π
at x = π
29 1, if
5
( )
3
5, if
5
kx
x
f x
x
x
+
≤
=
−
>
at x = 5
30 Find the values of a and b such that the function defined by
5,
if
2
( )
, if 2
10
21,
if
10
x
f x
ax
b
x
x
≤
=
+
<
<
≥
is a continuous function
|
1
|
2003-2006
|
1, if
( )
cos ,
if
kx
x
f x
x
x
+
≤ π
=
> π
at x = π
29 1, if
5
( )
3
5, if
5
kx
x
f x
x
x
+
≤
=
−
>
at x = 5
30 Find the values of a and b such that the function defined by
5,
if
2
( )
, if 2
10
21,
if
10
x
f x
ax
b
x
x
≤
=
+
<
<
≥
is a continuous function 31
|
1
|
2004-2007
|
1, if
5
( )
3
5, if
5
kx
x
f x
x
x
+
≤
=
−
>
at x = 5
30 Find the values of a and b such that the function defined by
5,
if
2
( )
, if 2
10
21,
if
10
x
f x
ax
b
x
x
≤
=
+
<
<
≥
is a continuous function 31 Show that the function defined by f(x) = cos (x2) is a continuous function
|
1
|
2005-2008
|
Find the values of a and b such that the function defined by
5,
if
2
( )
, if 2
10
21,
if
10
x
f x
ax
b
x
x
≤
=
+
<
<
≥
is a continuous function 31 Show that the function defined by f(x) = cos (x2) is a continuous function 32
|
1
|
2006-2009
|
31 Show that the function defined by f(x) = cos (x2) is a continuous function 32 Show that the function defined by f(x) = |cos x | is a continuous function
|
1
|
2007-2010
|
Show that the function defined by f(x) = cos (x2) is a continuous function 32 Show that the function defined by f(x) = |cos x | is a continuous function 33
|
1
|
2008-2011
|
32 Show that the function defined by f(x) = |cos x | is a continuous function 33 Examine that sin |x| is a continuous function
|
1
|
2009-2012
|
Show that the function defined by f(x) = |cos x | is a continuous function 33 Examine that sin |x| is a continuous function 34
|
1
|
2010-2013
|
33 Examine that sin |x| is a continuous function 34 Find all the points of discontinuity of f defined by f(x) = |x | – |x + 1 |
|
1
|
2011-2014
|
Examine that sin |x| is a continuous function 34 Find all the points of discontinuity of f defined by f(x) = |x | – |x + 1 | 5
|
1
|
2012-2015
|
34 Find all the points of discontinuity of f defined by f(x) = |x | – |x + 1 | 5 3
|
1
|
2013-2016
|
Find all the points of discontinuity of f defined by f(x) = |x | – |x + 1 | 5 3 Differentiability
Recall the following facts from previous class
|
1
|
2014-2017
|
5 3 Differentiability
Recall the following facts from previous class We had defined the derivative of a real
function as follows:
Suppose f is a real function and c is a point in its domain
|
1
|
2015-2018
|
3 Differentiability
Recall the following facts from previous class We had defined the derivative of a real
function as follows:
Suppose f is a real function and c is a point in its domain The derivative of f at c is
defined by
0
(
)
( )
lim
h
f c
h
f c
h
→
+
−
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
119
f (x)
xn
sin x
cos x
tan x
f ′(x)
nxn – 1
cos x
– sin x
sec2 x
provided this limit exists
|
1
|
2016-2019
|
Differentiability
Recall the following facts from previous class We had defined the derivative of a real
function as follows:
Suppose f is a real function and c is a point in its domain The derivative of f at c is
defined by
0
(
)
( )
lim
h
f c
h
f c
h
→
+
−
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
119
f (x)
xn
sin x
cos x
tan x
f ′(x)
nxn – 1
cos x
– sin x
sec2 x
provided this limit exists Derivative of f at c is denoted by f ′(c) or
( ( )) |c
d
dxf x
|
1
|
2017-2020
|
We had defined the derivative of a real
function as follows:
Suppose f is a real function and c is a point in its domain The derivative of f at c is
defined by
0
(
)
( )
lim
h
f c
h
f c
h
→
+
−
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
119
f (x)
xn
sin x
cos x
tan x
f ′(x)
nxn – 1
cos x
– sin x
sec2 x
provided this limit exists Derivative of f at c is denoted by f ′(c) or
( ( )) |c
d
dxf x The
function defined by
0
(
)
( )
( )
lim
h
f x
h
f x
f
x
h
→
+
−
′
=
wherever the limit exists is defined to be the derivative of f
|
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