Can you give an example of such a function? They very well may exist. But I think we can agree that non-differentiability of the target function is not sufficient.
The function that describes the neural network itself has to be differentiable. Whether we can create a differentiable function/NN for any kind of input remains to be shown.
Brownian motion. Try using a NN to model stock prices (not Brownian exactly, but same concept).
IIRC, the first neural network book I read in the 1990s had that as the big illustrative application in the latter part of the book.
There are plenty of other time series problems where NNs don't outperform classical methods such as ARIMA.
There is no empirical data to support that it is more effective than other methods.
Theory is not that valuable when evaluating papers in AI. It is all about the empirical results.