Let me try an answer and correct me if I am wrong.

First of all, we need that for every argument t the function converges to a fix f(t). But this is the case, otherwise f would not be a function, correct? Now, we may think of f(t) as a "superposition" of shifted Diracs weighted by the values f(t) at t. This summation must be infinitely dense, hence it becomes an integral.

But, no matter whether it is an integral or a sum, it means that we may convolve it with a Gaussian. A precondition for that is that our function f(t) does not grow faster than polynomials such that it is a tempered distribution. And because tempered distributions can be convolved with a Gaussian (which is a Schwartzfunction, see "On the Duality of Regular and Local Functions"), we end up with an infinitely smooth function after convolution.

Hence, the property of Gaussians which are infinitely smooth carries over to our previously nowhere differentiable function.

A direct approach can also be saying that, okay, if the sum(s) of a nowhere differentiable function is/are converging then we are allowed to convolve this function with another function and the convolution symbol can be drawn inside the sum(s). Using a Gaussian, the result will then be smooth (inifinitely differentiable).

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Hi,

It appears to be more of a definition or choise of averaging.  I think it may also refer to a kind of homogenisation. Sometimes it is a good description of such peaks, and in other cases, the peaks could  also be relevant.

In mixed fluid flow, while averaging(spatial) into a density, a volume fraction concept is used, which gives extended particles a larger density than this normal averaging would have done.

For time dependency, the relaxation between the peaks is important, as well as the nonsymmetry of time and a zero. 

Kind regards

This problem is of course related to the famous article of Mandelbrot on the length of the British coastline. It remains infinite as long as the coastline is a fractal but becomes finite if it is smoothed (approximated).

https://en.wikipedia.org/wiki/How_Long_Is_the_Coast_of_Britain%3F_Statistical_Self-Similarity_and_Fractional_Dimension 

Then it is spatial.

Or it is HaussDorf dimension with a width and a density ( that may be the subject of courses)

Kind regards