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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