I already know they do exist in signal processing, I have built many myself. It is a whole art to do so in any discipline that handles noisy data.
But is it possible to find out which nondifferentiable functions could be differentiable in such a sense?
What I am looking for is some macroscopification of the differential operator. For example something like:
$$D\{w,\Delta,f\}=\frac{1}{2\Delta}\int_{-\Delta}^{\Delta} w(t)\frac{f(t+\Delta)-f(t)}{\Delta}dt$$ where $w$ is a smooth positive weight-function and $\Delta > 0$.
For any positive $\Delta$ and continuous $f$ this should be well defined and the integral should exist, right?
Two famous examples from computer vision and image compression are Derivatives in Scale Spaces and Wavelets, in specific wavelets which are odd. Primitivest example is probably Haar wavelet. Slightly less primitive and more close to Fourier transform is imaginary part of Gabor wavelet if the band is not narrower and higher in frequency domain so it has more than circa 1 zero crossing.
Example of Gabor wavelet with odd and even part. However in signal processing application very often one tries to design so that mean value of the functions are 0. It is clear that the even/real part does not fulfill this in our example.