Question

For a given point on a function, does differentiability imply continuity?

Answer 1

Yes, if a function is differentiable in a point it is also continuous.

Explanation:

Suppose in fact `f(x)` defined in `(a,b)` and differentiable in the point `x_0 in (a,b)`

Then the limit:

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

exists and is finite.

Now because:

`lim_(x->x_0) (x-x_0) =0`

we must have also:

`lim_(x->x_0) (f(x)-f(x_0)) =0`

otherwise the limit could not be finite, but this implies:

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

which means that `f(x)` is continuous.