[mlthoughts2018]

It’s a little trickier than that, but you are generally right. The relaxation can involve many different things though, not just piecewise differentiability.

For example, when processing images that themselves may be intensity functions over a spatial domain, and the intensity function can have cusps, edges, occlusions, etc., then you need different weak-sense differentiability conditions, such as from Sobolev spaces, to guarantee that numerical gradient operators will succeed / converge where needed.

https://en.m.wikipedia.org/wiki/Sobolev_space