# What is the difference between differentiability and continuity of a function?

Feb 7, 2017

See explanation below

#### Explanation:

A function $f \left(x\right)$ is continuous in the point ${x}_{0}$ if the limit:

${\lim}_{x \to {x}_{0}} f \left(x\right)$

exists and is finite and equals the value of the function:

$f \left({x}_{0}\right) = {\lim}_{x \to {x}_{0}} f \left(x\right)$

A function $f \left(x\right)$ is differentiable in the point ${x}_{0}$ if the limit:

$f ' \left({x}_{0}\right) = {\lim}_{x \to {x}_{0}} \frac{f \left(x\right) - f \left({x}_{0}\right)}{x - {x}_{0}}$

exists and is finite.

A differentiable function is always continuous.
We can prove it by writing $f \left(x\right)$ as:

$f \left(x\right) = f \left({x}_{0}\right) + \frac{f \left(x\right) - f \left({x}_{0}\right)}{x - {x}_{0}} \left(x - {x}_{0}\right)$

Passing to the limit for $x \to {x}_{0}$:

${\lim}_{x \to {x}_{0}} f \left(x\right) = f \left({x}_{0}\right) + {\lim}_{x \to {x}_{0}} \left(\frac{f \left(x\right) - f \left({x}_{0}\right)}{x - {x}_{0}}\right) \cdot {\lim}_{x \to {x}_{0}} \left(x - {x}_{0}\right)$

${\lim}_{x \to {x}_{0}} f \left(x\right) = f \left({x}_{0}\right) + f ' \left({x}_{0}\right) \cdot 0 = f \left({x}_{0}\right)$

A function can be continuous but not differentiable, for example:

$y = \left\mid x \right\mid$

is continuous but not differentiable in $x = 0$.