Basic Idea:

A real function f is said to be continuous at x = c, where c is any point in the domain of f if :

where h is a very small ‘+ve’ no.

i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.

This is very precise, using our fundamental idea of the limit from class 11 we can summarise it as a function is continuous at x = c if :

Given :

……………….Equation 1

As for x ≠ 0, f(x) is just a product of two everywhere continuous function

∴ it is continuous for all x ≠ 0.

∵ f(x) is changing its nature at x = 0, So we need to check continuity at x = 0

f(0) = 0 [using equation 1]

and = 0

[∵ sin(1/0) is also going to be a value between [–1,1] ,so its product with 0 = 0]

Thus,

∴ It is continuous at x = 0

Hence, it is everywhere continuous.