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 :

NOTE:

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

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