Discuss the continuity of f(x) = sin |x|.

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 :


Here we are given with

f(x) = sin |x|

…………………………equation 1

Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain ( domain = set of numbers for which f is defined )

Function is changing its nature (or expression) at x = 0, So we need to check its continuity at x = 0 first.

LHL = =

[using eqn 1]

RHL = [using eqn 1]

f(0) =

[using eqn 1]

Clearly, LHL = RHL = f(0)

function is continuous at x = 0

For all x ≠ 0, f(x) is simply a trigonometric function which is everywhere continuous which can be verified by seeing its plot in X–Y plane or even can be verified using limit.

f(x) is everywhere continuous in its domain i.e. it is continuous for all real values of x