# Differentiable function

### From Conservapedia

(Redirected from Differentiability)

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 ε − δ definition: a function f is said to be differntiable at point a if ∀ε > 0 ∃​δ > 0 such that if

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

then

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