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On Wikipedia, it says that result of differentiating under the integral sign is essentially the same as the result of Schwarz's theorem of the symmetry of partial derivatives. My problem is that in the latter we need the function to be in the class $C^2$, while in the former we only need it to be in $C^1$. What's the reason for the contradiction?

I mean, if a certain function is in $C^1$ (but not in $C^2$, so that partial derivatives don't commute), can I still differentiate under the integral sign?

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