Find the points of discontinuity, if any, of the following functions :

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 limit from class 11 we can summarise it as, A function is continuous at x = c if :

Here we have,

……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 )

Let c is any random number such that c < 1 [thus c being random number, it is able to include all numbers less than 1]

f(c) = [ using eqn 1]

Clearly,

∴ We can say that f(x) is continuous for all x < 1

As x = 1 is a point at which function is changing its nature so we need to check the continuity here.

f(1) = 1¹⁰ = 1 [using eqn 1]

LHL =

RHL =

Thus LHL = RHL = f(1)

∴ f(x) is continuous at x = 1

Let m is any random number such that m > 1 [thus m being random number, it is able to include all numbers greater than 1]

f(m) = [ using eqn 1]

Clearly,

∴ We can say that f(x) is continuous for all m > 1

Hence, f(x) is continuous for all real x

There no point of discontinuity. It is everywhere continuous