It is within the context of distributions or generalized functions (https://en.wikipedia.org/wiki/Distribution_(mathematics)) but people are often loose on the terminology and tend to just use the term "functions". It's a wonderful topic, with a lot of interesting applications in differential equations and physics.

I just found a quick explanation by Terence Tao about why people are generally loose in this case, meaning that some properties transition nicely from smooth (here, differentiable) top the rough categories by passing to the limit and density arguments: http://www.math.ucla.edu/~tao/preprints/distribution.pdf

Of course there are exceptions.