Examine the continuity of f, where f is defined by f(x)={(sin x − cos x, if x≠0) (−1, if x = 0)

Solution:

The given function is

f(x) = {sin x − cos x), if x ≠ 0 − 1, if x = 0

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 ≠ 0,

then f(c) = sin c − cos c 

lim_x→c f(x) = lim_x→c (sin x − cos x)

= sin c − cos c

⇒ lim_x→c f(x) = f(c)

Therefore, f is continuous at all points x, such that x ≠ 0 .

Case II:

If c = 0 , then f(0) = −1

lim_x→0⁻ f(x) = lim_x→0 (sin x − cos x)

= sin 0 − cos 0

= 0 − 1 = −1

lim_x→0⁺ f(x) = lim_x→0 (sin x − cos x)

= sin 0 − cos 0

= 0 − 1 = −1

lim_x→0⁻ f(x) = lim_x→0⁺ f(x) = f(0)

Therefore, f is continuous at x = 0.

From the above observations, it can be concluded that f is continuous at every point of the real line.

Thus, f is a continuous function

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 25

Examine the continuity of f, where f is defined by f(x)={(sin x − cos x, if x≠0) (−1, if x=0)

Summary:

For the function defined by f(x)= (sin x − cos x, if x ≠ 0) (−1, if x = 0) it can be concluded that f is continuous at every point of the real line. Thus, f is a continuous function