Find all points of discontinuity of f, where f is defined by f(x) = {(x/|x|, if x < 0) (−1, if x ≥ 0)

Solution:

A function is said to be continuous when the graph of the function is a single unbroken curve.

The given function is 

f(x) = {(x/|x|, if x < 0) (−1, if x ≥ 0)

It is known that x < 0

⇒ |x| = −x

Therefore, the given function can be rewritten as

f(x)={(x/|x| = x/−x = −1, if x < 0) (−1, if x ≥ 0)

⇒ f(x) = −1∀ x ∈ R

∀ represents 'for all' and ∈ represents 'belongs to'

Let c be any real number.

Then, 

lim_x→c f(x) = lim_x→c (−1)

= −1

Also,

f(c) = −1 = lim_x→c f(x)

Therefore, the given function is a continuous function.

Hence, the given function has no point of discontinuity

NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 9

Find all points of discontinuity of f, where f is defined by f(x) = {(x/|x|, if x < 0) (−1, if x ≥ 0)

Summary:

For the function f defined by f(x) = {(x/|x|, if x < 0) (−1, if x ≥ 0), is a continuous function. Hence, the given function has no point of discontinuity