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I am interested to know what topological properties are possessed by non-smooth functions. I suspect this is well known, but I can't find what I am looking for from google.

Is there a topological "definition" of non-smooth functions (something elegant like the open set def of continuous functions) ?

My motivation involves the Naiver-Stokes millennium problem.

My intuition tells me that the highest derivative of a non-smooth function should be a non-differentiable continuous function, and furthermore that the non-differentiability is reflected in some form of scaling "symmetry", actually I suspect there is a broken scaling symmetry (i.e. the non-smooth solutions of N-S violate the scaling symmetry of the N-S equations, while still maintaining some sort of correlation across length scales)

The basic idea would be to show that the broken scale symmetry of the solutions of N-S have a topological property which forces them to be non-smooth. I'm certain this approach has massive issue's that need to be resolved, and that it has been thought of before since it's kind of obvious. Does anyone know about any results in this direction?

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(This is not really intended as an answer, but was too long to fit as a comment)

You're very unlikely to find what you're looking for. Non-smooth is just that, not smooth. So a non-smooth function is just one that doesn't have infinitely many derivatives. If you assume continuous non-smooth functions, then you have precisely the open set definition. If you want some derivatives, then you're looking at $C^k$ functions, which have $k$ continuous derivatives. These have various properties, but smoothness and differentiability depend on structure beyond topological. As for "scaling" symmetry, what exactly do you mean by that? There's a scaling "symmetry" for homogeneous functions of a given degree, by $f(\lambda x)=\lambda^d f(x)$, but nothing of the sort really for an arbitrary function, smooth or nonsmooth.

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  • $\begingroup$ It's also worth saying that we often use "non-smooth" to mean "not necessarily smooth". So a "non-smooth function" can actually be smooth! See ncatlab.org/nlab/show/red+herring+principle for more fun on this. $\endgroup$
    – Andrew Stacey
    Commented Aug 19, 2010 at 19:08
  • $\begingroup$ @Charlie: Which structure's beyond topology do smoothness and differentiability depend on? I guess I want to know if there is any way to tell if a function is not smooth without just trying to take derivatives of every order. By scaling symmetry I mean for a differential equation. Check out this post by Terry Tao for the scaling symmetry of N-S. I guess I just feel like if a function has a fractal type scaling construction, it can't be smooth (because the derivatives of some order will oscillate) $\endgroup$
    – Matt Calhoun
    Commented Aug 19, 2010 at 19:50
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    $\begingroup$ It depends on a differentiable structure. See en.wikipedia.org/wiki/Differentiable_structure and mathoverflow.net/questions/24970/… $\endgroup$
    – Charles Siegel
    Commented Aug 19, 2010 at 20:22

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