Basic Concept:

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 = 1, So we need to check its continuity at x = 1 first.

Clearly,

f(1) = 4 [using eqn 1]

Clearly,

∴ f(x) is discontinuous at x = 1.

Let c be any real number such that c ≠ 0

f(c) = c³ – c² + 2c – 2 [using eqn 1]

Clearly,

∴ f(x) is continuous for all real x except x =1