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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?

Joe Corneli
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  • I think we are fine as long as the singular points form a set of measure zero. Short of that, we are in trouble, and it is about as bad as it looks. – Ivan Neretin Feb 20 '19 at 15:41
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    What is your definition of solution to an ODE? – Romeo Feb 20 '19 at 16:37
  • @Romeo not sure yet! But, e.g., in this case perhaps it's relevant to work with the Laplace transform to generate a "weak solution". – Joe Corneli Feb 20 '19 at 17:00

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An ODE has at least one derivative expression in it. A function $f$ that is differentiable almost nowhere is going to be undefined in the ODE almost everywhere. So it isn't logically possible for an $f$ undefined almost everywhere in an equation to be a solution to said equation.

Fomalhaut
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    Logic does not always apply in mathematics in the way you would expect. What is the sum of 1+2+3+4+... ? – Joe Corneli Feb 20 '19 at 17:07
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    “Sum” in that sense is not the same as the usual “sum” you’re thinking of. The series $\sum_{n=1}^\infty n$ indeed is divergent in the usual sense of the word sum. – Curious Feb 20 '19 at 17:12
  • I am similarly happy for "derivative" to be reinterpreted in the definition of ODEs. Consider e.g. solutions in the sense of distributions. – Joe Corneli Feb 20 '19 at 17:33