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.



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> **Q.** Is there some nice equivalent condition to the fact that a real function has antiderivates?
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Equivalent conditions for a real function to have antiderivates
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.


