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Lebesgue theorem says that a bounded function $f$ is Riemann Integerable if and only if $f$ continuous almost everywhere.

Unfortunately, we know a function has antiderivative has no relation to Riemann Integerable.

Q. Is there some nice equivalent condition to the fact that a real function has antiderivates?

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There are various characterizations of real functions which have antiderivatives. See, for example, Theorem 4 in M. W. Botsko: Exactly which bounded Darboux functions are derivatives?, Amer. Math. Monthly 114 (2007), 242-246.

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