# Find all points of discontinuity of f, where f is defined by f(x) = {(x + 1, if x ≥ 1) (x^{2 }+ 1, if x < 1)

**Solution:**

The given function is f(x) = {(x + 1, if x ≥ 1) (x^{2 }+ 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) = c^{2} + 1

lim_{x→c} f(x) = lim_{x→c} (x^{2 }+ 1)

= c^{2} + 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−} (x^{2 }+ 1)

= 1^{2} + 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

## Find all points of discontinuity of f, where f is defined by f(x) = {(x + 1, if x ≥ 1) (x^{2 }+ 1, if x < 1)

**Summary:**

For the function f defined by f(x) = {(x + 1, if x ≥ 1) (x^{2 }+ 1, if x < 1), f is continuous at all points x, such that x > 1. Hence, the given function f has no point of discontinuity