Chapter 5 Continuity and Differentiability Ex 5.1 Solutions
Question - 1 : - Prove that the function
is continuous at
Answer - 1 : -
Therefore, f iscontinuous at x = 0
Therefore, f iscontinuous at x = −3
Therefore, f iscontinuous at x = 5
Answer - 2 : -
Thus, f iscontinuous at x = 3
Examine the following functions forcontinuity.
(a)
(b)
(c) 
(d) 
Answer - 3 : - (a) The given function is
It isevident that f is defined at every real number k andits value at k is k − 5.
It is also observedthat, 
Hence, f iscontinuous at every real number and therefore, it is a continuous function.
(b) The givenfunction isForany real number k ≠ 5, we obtain
Hence, f iscontinuous at every point in the domain of f and therefore, itis a continuous function.
(c) The givenfunction is
Forany real number c ≠ −5, we obtain
Hence, f iscontinuous at every point in the domain of f and therefore, itis a continuous function.
(d) The given function is 
Thisfunction f is defined at all points of the real line.
Let c bea point on a real line. Then, c < 5 or c =5 or c > 5
CaseI: c < 5
Then, f (c)= 5 − c
Therefore, f iscontinuous at all real numbers less than 5.
CaseII : c = 5
Then, 
Therefore, f iscontinuous at x = 5
Case III: c >5
Therefore, f iscontinuous at all real numbers greater than 5.
Hence, f iscontinuous at every real number and therefore, it is a continuous function.
Question - 4 : - Prove that the function 
is continuous at x = n, where n isa positive integer.
Answer - 4 : -
Thegiven function is f (x) = xn
It isevident that f is defined at all positive integers, n,and its value at n is nn.
Therefore, f iscontinuous at n, where n is a positive integer.
Question - 5 : - Isthe function f defined by
continuousat x = 0? At x = 1? At x =2?
Answer - 5 : - The given function f is
At x =0,
It isevident that f is defined at 0 and its value at 0 is 0.
Therefore, f iscontinuous at x = 0
At x =1,
f is defined at 1 andits value at 1 is 1.
Theleft hand limit of f at x = 1 is,
Theright hand limit of f at x = 1 is,
Therefore, f isnot continuous at x = 1
At x =2,
f is defined at 2 andits value at 2 is 5.
Therefore, f iscontinuous at x = 2
Findall points of discontinuity of f, where f is definedby
Answer - 6 : - The given function f is
It isevident that the given function f is defined at all the pointsof the real line.
Let c bea point on the real line. Then, three cases arise.
(i) c <2
(ii) c >2
(iii) c =2
Case (i) c <2Therefore, f iscontinuous at all points x, such that x < 2
Case (ii) c >2Therefore, f iscontinuous at all points x, such that x > 2
Case(iii) c = 2
Then, the left handlimit of f at x = 2 is,Theright hand limit of f at x = 2 is,
It isobserved that the left and right hand limit of f at x =2 do not coincide.
Therefore, f isnot continuous at x = 2
Hence, x =2 is the only point of discontinuity of f.
Findall points of discontinuity of f, where f isdefined by
Answer - 7 : - The given function f is
Thegiven function f is defined at all the points of the realline.
Let c bea point on the real line.
Case I:Therefore, f iscontinuous at all points x, such that x < −3
Case II:
Therefore, f iscontinuous at x = −3
Case III:
Therefore, f iscontinuous in (−3, 3).
Case IV:
If c =3, then the left hand limit of f at x = 3 is,
Theright hand limit of f at x = 3 is,
It isobserved that the left and right hand limit of f at x =3 do not coincide.
Therefore, f isnot continuous at x = 3
Case V:
Therefore, f iscontinuous at all points x, such that x > 3
Hence, x =3 is the only point of discontinuity of f.
Findall points of discontinuity of f, where f isdefined by
Answer - 8 : - The given function f is
It is known that,
Therefore, the given function can berewritten as
Thegiven function f is defined at all the points of the realline.
Let c bea point on the real line.
Case I:
Therefore, f iscontinuous at all points x < 0
Case II:
If c =0, then the left hand limit of f at x = 0 is,
Theright hand limit of f at x = 0 is,
It isobserved that the left and right hand limit of f at x =0 do not coincide.
Therefore, f isnot continuous at x = 0
Case III:
Therefore, f iscontinuous at all points x, such that x > 0
Hence, x =0 is the only point of discontinuity of f.
Findall points of discontinuity of f, where f isdefined by
Answer - 9 : - The given function f is
It is known that,
Therefore, the given function can berewritten as
Let c be any real number. Then, 
Also,
Therefore, the given function is a continuousfunction.
Hence, the givenfunction has no point of discontinuity.
Findall points of discontinuity of f, where f isdefined by
Answer - 10 : - The given function f is
Thegiven function f is defined at all the points of the realline.
Let c bea point on the real line.
Case I:
Therefore, f iscontinuous at all points x, such that x < 1
Case II:
Theleft hand limit of f at x = 1 is,
Theright hand limit of f at x = 1 is,
Therefore, f iscontinuous at x = 1
Case III:
Therefore, f iscontinuous at all points x, such that x > 1
Hence,the given function f has no point of discontinuity.