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 :

NOTE: If f and g are two functions whose domains are same and both f and g are everywhere continuous then f/g is also everywhere continuous for all R except point at which g(x) = 0

As, f(x) =

Domain of f = { all Real numbers except 2 } = R – {–2}

Clearly it is not defined at x = –2, for rest of values it is continuous everywhere

Because 1 is everywhere continuous and x + 2 is also everywhere continuous

∴ f(x) is everywhere continuous except at x = –2

f(f(x)) = f

Domain of f(f(x)) = R – {–2 , (}

For rest of values it just a fraction of two everywhere continuous function

∴ at all other points it is everywhere continuous.

Hence,

f(f(x)) is discontinuous at x = –2 and x = –5/2