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> Cantor's function shows there are functions that are not the integrals of their derivatives. That's because Cantor's function does not have a derivative that
by yorwba 6y ago
> Cantor's function shows there are functions that are not the integrals of their derivatives.
That's because Cantor's function does not have a derivative that's defined everywhere. Of course a function without derivative is not the integral of that non-existent derivative.
If most people think that increasing/monotone functions must have positive derivatives, they're apparently forgetting about non-differentiable functions.
- memexy 6y agoThe function is differentiable almost everywhere (this is a technical term from measure theory). Here's a pretty good explanation of its properties: https://epicmath.org/2012/11/27/5-the-cantor-set-and-the-cantor-function/ https://epicmath.org/2012/11/27/5-the-cantor-set-and-the-can.... > The Cantor function, also known as the Cantor Staircase, is a bizarre function that is continuous and has a derivative of 0 at every point where it is differentiable. In fact, it is differentiable at every point other than on the Cantor set, which is a set of measure zero. So from a measure theoretic perspective the derivative of the Cantor function is well defined and it is equal to 0 almost everywhere. Almost everywhere equality is an equivalence relation and the derivative of the Cantor function is in the same equivalence class as the 0 function.
- yorwba 6y ago> Almost everywhere equality is an equivalence relation and the derivative of the Cantor function is in the same equivalence class as the 0 function. And a function in that equivalence class will have an integral that is equal to a constant function on all intervals where that function is defined. If you also integrate over intervals where the function is only defined almost everywhere, all bets are off.