Question

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

Answer 1

See explanation below

Explanation:

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`.