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

Question

8540 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-02-07T10:14:17

See explanation below

A function $f(x)$ is continuous in the point $x_0$ if the limit:

$lim_(x->x_0) f(x)$

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

$f(x_0) = lim_(x->x_0) f(x)$

A function $f(x)$ is differentiable in the point $x_0$ if the limit:

$f'(x_0) = lim_(x->x_0) (f(x)-f(x_0))/(x-x_0)$

exists and is finite.

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

$f(x) = f(x_0) + (f(x) - f(x_0))/(x-x_0)(x-x_0)$

Passing to the limit for $x->x_0$ :

$lim_(x->x_0) f(x) = f(x_0) + lim_(x->x_0) ((f(x) - f(x_0))/(x-x_0))*lim_(x->x_0) (x-x_0)$

$lim_(x->x_0) f(x) = f(x_0) + f'(x_0)*0= f(x_0)$

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

$y = abs(x)$

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