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In Lebesgue integration, if you change a function on a countable subset of its domain, neither integrability nor the value of the integral changes. The same is obviously not true for differentiation, which is locally defined through a limit.

Now I wonder: Is there a definition of differentiation that also is "immune" against changes on countable subsets?

Well, I guess one could simply define "a function $f$ is X-differentiable in $x$ if there exists a function $g$ that is differentiable in $x$ and agrees with $f$ almost everywhere", but that's sounds like cheating.

So actually my question is: Does there exist a natural definition of derivative with this property? Where "natural" means that the definition makes sense even if you don't know about the property that it should be "immune" against changes on countable sets.

celtschk
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2 Answers2

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Yes, the Lanczos generalized derivative will have that
property whenever its integration has that property.

4

Look up "absolutely continuous functions", " Sobolev spaces", "weak derivative" and "distributions". The approaches that I know of are based on on one of two things: either you tell that one object (maybe not even function, e.g. distributions) is derivative of other if integration by parts " works as usually". The other one is based on noting that non differentiable (in usual sense) function $f$ can be integral of some function $g$; then it's natural to write $g=f'$. In some cases those approaches turn out to be equivalent. By the way, physicists use those derivatives all the time, usually not even thinking much because it comes up so naturally and often in practice and works really well.

Blazej
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