Is the function f defined by f(x) = {(x, if x ≤ 1), (5, if x > 1) continuous at x = 0? At x = 1? At x = 2?

Solution:

The given function is

f(x) = {(x, if x ≤ 1) (5, if x > 1)

At x = 0,

It is evident that f is defined at 0 and its value at 0 is 0.

Then,

lim_x→0 f(x) = lim_x→0 (x) = 0

⇒ lim_x→0 f(x) = f(0)

Therefore, f is continuous at x = 0.

At x = 1,

It is evident that f is defined at 1 and its value at 1 is 1.

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

lim_x→1⁻ f(x )= lim_x→1⁻ (x) = 1

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

lim_x→1⁺ f(x) = lim_x→1⁺ (5) = 5

⇒ lim_x→1⁻ f(x) ≠ lim_x→1⁺ f(x)

Therefore, f is not continuous at x = 1.

At x = 2,

It is evident that f is defined at 2 and its value at 2 is 5.

lim_x→2 f(x) = lim_x→2 (5) = 5

⇒ lim_x→1 f(x) = f(2)

Therefore, f is continuous at x = 2

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

Is the function f defined by f(x) = {(x, if x ≤ 1), (5, if x > 1) continuous at x = 0? At x = 1? At x = 2?

Summary:

For the given function f  defined by f(x) = {(x, if x ≤ 1), (5, if x > 1)is continuous at x = 0.f is not continuous at x = 1.f is continuous at x = 2