Pointwise derivative Please I am struggling to find a proper definition of a pointwise derivative and also what is the difference between a pointwise derivative and the "classical" derivative.
Can someone please point me to a good reference or define it themselves.
Thank you in advance. 
 A: 
Function $f : [a, b] \to\mathbb  R $ is absolutely continuous if and only if: (a) the pointwise derivative $f'$ exists almost everywhere in $(a, b)$
...why pointwise and not the standard derivative?

The pointwise derivative is the "standard" derivative, $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$. There are a few reasons to attach the label pointwise to it in this context:

*

*Preparation for introduction of weak derivatives, which is another concept of derivative. In some sense, the weak derivative is the "right" concept, as it is exactly what the fundamental theorem of calculus needs. From this point of view, the "standard" derivative is inferior: it captures  the behavior of  $f$ in a particular point, but sometimes fails to show how the function behaves on large scale.  The adjective pointwise aptly labels this flaw.

*It reminds the reader that $f'(x)$ is something that exists on a point-to-point basis, not necessarily everywhere

*It separates the concept from classical differentiability spaces such as $C^1$ , where the derivative is assumed not only to exist everywhere, but to also be continuous. For example, in PDE theory a "classical solution"  refers to a solution in some $C^k$ space.

