Definition:Differentiable Mapping

At a Point
Let $$f$$ be a real function defined on an open interval $$I$$.

Let $$\xi \in I$$ be a point in $$I$$.

Then $$f$$ is differentiable at the point $$\xi$$ iff the limit $$\lim_{x \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$$ exists.

On an Interval
Let $$f$$ be a real function defined on an open interval $$I$$.

Let $$f$$ be differentiable at each point of $$I$$.

Then $$f$$ is differentiable on $$I$$.