##### Discuss the continuity of the function .

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 have, …….equation 1

The 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.

NOTE: LHL = = = [using eqn 1 and idea of mod fn]

RHL = [using eqn 1 and idea of mod fn]

f(0) = 0

[using eqn 1]

Clearly, LHL ≠ RHL ≠ f(0)

function is discontinuous at x = 0

Let c be any real number such that c > 0

f(c) = [using eqn 1]

And, Thus, f(x) is continuous everywhere for x > 0.

Let c be any real number such that c < 0

f(c) = [using eqn 1 and idea of mod fn]

And, Thus, f(x) is continuous everywhere for x < 0.

Hence, We can conclude by stating that f(x) is continuous for all Real numbers except zero.

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