A different interpretation of your question:
For what distribution is the distributional derivative the same as the elementary derivative of a given function?
Observe that in the above formula for the distributional derivative, if the mapping f is linear, then T is the same as f. (The...
The distributional derivative can be defined in terms of a Fréchet derivative on general normed linear spaces.
For a mapping f from one normed linear space to another, the derivative T at an element x in the domain is the linear operator with the same domain and range that satisfies
limδ→0...