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.

This limit, if it exists, is called the derivative of $$f$$ at $$\xi$$.

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 $$I$$.

On the Real Number Line
In the definition of differentiable on an interval let that interval be the real number line $$\R$$.

Let $$f$$ be differentiable at each point of $$\R$$.

Then $$f$$ is differentiable everywhere (on $$\R$$).

In the Complex Plane
Let $$f \left({z}\right): \C \to \C \ $$ be a single-valued continuous complex function in a domain $$D \subseteq \C$$.

Let $$z_0 \in D$$ be a point in $$D$$.

Then $$f \left({z}\right)$$ is complex-differentiable at $$z_0 \ $$ iff the limit $$\lim_{h \to 0} \frac {f \left({z_0+h}\right) - f \left({z_0}\right)} {h}$$ exists as a finite number and is independent of how the complex increment $$h \ $$ tends to $$0 \ $$.

If such a limit exists, it is called the derivative of $$f \ $$ at $$z_0 \ $$.

If $$f \left({z}\right)$$ is complex-differentiable at every point in $$D \ $$, it is differentiable in $$D \ $$. Such a function is called analytic.