Discuss the continuity of the following functions at the indicated point(s).

at x = 0

In this problem we need to check continuity at x = 0

Given function is

at x = 0 …… Equation 2

∴ we need to check LHL, RHL and value of function at x = 0 (for idea and meaning of continuity refer to Q10(i) )

Clearly,

f(0) = 0 [from equation 2]

LHL = [∵ sin(–θ) = sinθ]

RHL = [ using eqn 2]

Why

Since whatever is value of h, sin(1/h) is going to range from –1 to 1

As h→ 0 i.e. h is approximately 0

∴ 0*some finite quantity is equal to 0.

Clearly, LHL = RHL = f(0)

∴ f(x) is continuous at x = 0

10