Chapter 5 Continuity and Differentiability Ex 5.1 Solutions
Question - 11 : - Find allpoints of discontinuity of f, where f is definedby
Answer - 11 : - 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 < 2
Case II:
Therefore, f iscontinuous at x = 2
Case III:
Therefore, f iscontinuous at all points x, such that x > 2
Thus,the given function f is continuous at every point on the realline.
Hence, f hasno point of discontinuity.
Findall points of discontinuity of f, where f isdefined by
Answer - 12 : - 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:
If c =1, then the left hand limit of f at x = 1 is,
Theright hand limit of f at x = 1 is,
It isobserved that the left and right hand limit of f at x =1 do not coincide.
Therefore, f isnot continuous at x = 1
Case III:
Therefore, f iscontinuous at all points x, such that x > 1
Thus,from the above observation, it can be concluded that x = 1 isthe only point of discontinuity of f.
Is the function defined by
a continuousfunction?
Answer - 13 : - The given function 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,
It isobserved that the left and right hand limit of f at x =1 do not coincide.
Therefore, f isnot continuous at x = 1
Case III:
Therefore, f iscontinuous at all points x, such that x > 1
Thus,from the above observation, it can be concluded that x = 1 isthe only point of discontinuity of f.
Discussthe continuity of the function f, where f isdefined by
Answer - 14 : - The given function is
The given function is defined at all pointsof the interval [0, 10].
Let c bea point in the interval [0, 10].
Case I:
Therefore, f iscontinuous in the interval [0, 1).
Case II:
Theleft hand limit of f at x = 1 is,
Theright hand limit of f at x = 1 is,
It isobserved that the left and right hand limits of f at x =1 do not coincide.
Therefore, f isnot continuous at x = 1
Case III:
Therefore, f iscontinuous at all points of the interval (1, 3).
Case IV:
Theleft 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 limits of f at x =3 do not coincide.
Therefore, f isnot continuous at x = 3
Case V:
Therefore, f iscontinuous at all points of the interval (3, 10].
Hence, f isnot continuous at x = 1 and x = 3
Discussthe continuity of the function f, where f isdefined by
Answer - 15 : - The given function is
The given function is defined at all pointsof the real line.
Let c bea point on the real line.
Case I:
Therefore, f iscontinuous at all points x, such that x < 0
Case II:
Theleft hand limit of f at x = 0 is,
Theright hand limit of f at x = 0 is,
Therefore, f iscontinuous at x = 0
Case III:
Therefore, f iscontinuous at all points of the interval (0, 1).
Case IV:
Theleft hand limit of f at x = 1 is,
Theright hand limit of f at x = 1 is,
It isobserved that the left and right hand limits of f at x =1 do not coincide.
Therefore, f isnot continuous at x = 1
Case V:
Therefore, f iscontinuous at all points x, such that x > 1
Hence, f isnot continuous only at x = 1
Discussthe continuity of the function f, where f isdefined by
Answer - 16 : - The given function f is
The given function is defined at all pointsof the real line.
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 of the interval (−1, 1).
Case IV:
Theleft hand limit of f at x = 1 is,
Theright hand limit of f at x = 1 is,
Therefore, f iscontinuous at x = 2
Case V:
Therefore, f iscontinuous at all points x, such that x > 1
Thus,from the above observations, it can be concluded that f iscontinuous at all points of the real line.
Findthe relationship between a and b so that thefunction f defined by
iscontinuous at x = 3.
Answer - 17 : - The given function f is
If f iscontinuous at x = 3, then
Therefore, from (1), we obtain
Therefore, the required relationship is given by,
For what value of 
is the function defined by
continuousat x = 0? What about continuity at x = 1?
Answer - 18 : -
The given function f is If f iscontinuous at x = 0, then
Therefore,there is no value of λ for which f is continuous at x =0
At x =1,
f (1) = 4x +1 = 4 × 1 + 1 = 5
Therefore,for any values of λ, f is continuous at x = 1
is discontinuous at all integral point. Here 
denotesthe greatest integer less than or equal to x.
Answer - 19 : - The given function is
It isevident that g is defined at all integral points.
Let n bean integer.
Then,
Theleft hand limit of f at x = n is,
Theright hand limit of f at x = n is,
It isobserved that the left and right hand limits of f at x = n donot coincide.
Therefore, f isnot continuous at x = n
Hence, g isdiscontinuous at all integral points.
continuousat x = π?
Answer - 20 : - The given function is
It isevident that f is defined at x = π.
Therefore,the given function f is continuous at x = π