Suppose we have a first order Ordinary Differential Equation, $y^\prime(x)=f(x,y)$. On the face of it, it looks as though any solution $y$ should be differentiable throughout its domain, but this may not be the case if the function has singular points.
The question linked here constructs a differentiable function whose derivative has a Cantor set of discontinuities: Discontinuous derivative. Thinking along these lines, is it possible to go further and build an "ODE solution" that, like the Weierstrass function, is continuous but (almost ?) nowhere differentiable?
Notice that the antiderivative of the Weierstrass function is "fairly smooth", per What does the antiderivative of a continuous-but-nowhere-differentiable function "look like"?, so while $y=\int W$ solves $y^\prime=W(x)$, such an ODE does not have the weird property I'm asking about.
However, I wonder if $W$ itself, despite not being differentiable, could in some degenerate sense be considered an ODE solution?
The example of $\int W$ would appear to give circumstantial evidence to support the idea that integration is a smoothing operation, and that a solution to an ODE should therefore be mostly smooth. The only problem with that theory, of course, is that equations with singular solutions appear all over the place.
If any such bizarre functions could be constructed, are they forbidden on aesthetic or practical grounds in the theory of ODEs?