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

Note: [for changing the expression used identity:– (a^{2}–b^{2}) = (a+b)(a–b)]

Note: x – 2 is cancelled from numerator and denominator only because x ≠ 2, else we can’t cancel them

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 = 2, So we need to check its continuity at x = 2 first.

Clearly,

f(2) = 16 [using eqn 1]

Note: (x – 2) is cancelled as x ≠ 2 but x → 2

Clearly,

∴ f(x) is continuous at x = 2.

Let c be any real number such that c ≠ 0

f(c) = [using eqn 1]

Clearly,

∴ f(x) is continuous for all real x

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