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,
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) = \(\frac{1}{x+2}\)
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\((\frac{1}{x+2})\)
= \(\frac{1}{\frac{1}{x+2}+2}\)
= \((\frac{x+2}{2x+5})\)
Domain of f(f(x)) = R – {–2 , \((\frac{-5}{2})\)}
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
