Determine if f defined by f(x) = {(x² sin1/x, if x≠0) (0, if x=0) is a continuous function?

Solution:

The given function is 

f(x) = {(x² sin1/x, if x ≠ 0) (0, 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) = c² sin 1/c 

lim_x→c f(x) = lim_x→c (x² sin1/x)

= (lim_x→c x²) (lim_x→c sin1/x)

= c²sin 1/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) = 0

lim_x→0⁻ f(x) = lim_x→0⁻ (x² sin1/x)

= lim_x→0 (x² sin1/x) 

It is known that, −1 ≤ sin1/x ≤ 1, x ≠ 0

⇒ −x² ≤ x² sin1/x ≤ x²

⇒ lim_x→0 (−x²) ≤ lim_x→0 (x² sin1/x) ≤ limx_2x→0

⇒ 0 ≤ lim_x→0 (x² sin1/x) ≤ 0

⇒ lim_x→0 (x² sin1/x) = 0

⇒ lim_x→0⁻ f(x) = 0

Similarly,

lim_x→0⁺ f(x) = lim_x→0⁺ (x² sin1/x)

= lim_x→0 (x² sin1/x) = 0

⇒ lim_x→0⁻ f(x) = f(0) = lim_x→0⁺ f(x)

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 24

Determine if f defined by f(x) = {(x² sin1/x, if x≠0) (0, if x=0) is a continuous function?

Summary:

Function defined by f(x) = {(x² sin1/x, if x ≠ 0) (0, if x = 0) is a continuous function. From the above observations, it can be concluded that f is continuous at every point of the real line