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:

<math>\frac{f(x) - f(a)}{(x - a)}</math>

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

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

<math> |x - a| < \delta\,</math>

then

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