Find all points of discontinuity of f, where f is defined by f(x) = {(x + 1, if x ≥ 1) (x²+ 1, if x < 1)

Solution:

The given function is f(x) = {(x + 1, if x ≥ 1) (x²+ 1, if x < 1)

The given function f is defined at all the points of the real line.

Let c be a point on the real line.

Case I:

If c < 1, then f(c) = c² + 1

lim_x→c f(x) = lim_x→c (x²+ 1)

= c² + 1

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x, such that x < 1.

Case II:

If c = 1,

then f(c) = f(1)

= 1 + 1 = 2

The left hand limit of f at x = 1 is,

lim_x→1⁻ f(x) = lim_x→1⁻ (x²+ 1)

= 1² + 1 = 2

The right hand limit of f at x = 1 is,

lim_x→1⁺f(x) = lim_x→1⁺ (x + 1)

= 1 + 1 =2

⇒ lim_x→1 f(x) = f(1)

Therefore, f is continuous at x = 1

Case III:

If c > 1,

then f(c) = c + 1

lim_x→c f(x) = lim_x→c (x+1)

= c + 1

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x, such that x > 1.

Hence, the given function f has no point of discontinuity

NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 10