Differentiable function

A function f(x) is differentiable at the point a if and only if, as x approaches a (which it is never allowed to reach), the value of the quotient:

${\frac {f(x)-f(a)}{(x-a)}}$

approaches a limiting value that we call the derivative of the function f(x) at x=a.

There is also the more rigorous $\epsilon -\delta$ definition: a function f is said to be differntiable at point a if ∀$\epsilon >0$ ∃​$\delta >0$ such that if

$|x-a|<\delta \,$

then

$|{\frac {f(x)-f(a)}{x-a}}-f'(a)|<\epsilon \,$.