What do you mean by "good approximation"? Arbitrarily close?
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.
Once again I ask: What do you mean with "works well". Just because some other ML method is better in practice doesn't mean that a NN can not achieve the same degree of approximation in theory.
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.