Discuss the continuity of the function

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 )

Function is changing its nature (or expression) at x = 2, So we need to check its continuity at x = 2 first.

LHL = = = [using eqn 1]

RHL = [using eqn 1]

f(2) =

[using eqn 1]

Clearly, LHL = RHL = f(2)

∴ function is continuous at x = 2

Let c be any real number such that c > 2

∴ f(c) = [using eqn 1]

And,

Thus,

∴ f(x) is continuous everywhere for x > 2.

Let m be any real number such that m < 2

∴ f(m) = [using eqn 1]

And,

Thus,

∴ f(x) is continuous everywhere for x < 2.

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

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