We know that continuous functions
- are Riemann integrable and
- have an antiderivative.
Are there any other (hopefully huge :-) function classes fulfilling both of these properties?
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By Darboux theorem, any such function has to obey the intermediate value property. I am not certain whether this goes the other way though. A good example is the discontinuous $2x\sin(1/x)-\cos(1/x)$ which has antiderivative $x^2\sin(1/x)$. Both should equal $0$ at $x=0$. – SmileyCraft Oct 17 '18 at 23:07
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Also interesting: https://math.stackexchange.com/questions/257069/what-is-an-example-that-a-function-is-differentiable-but-derivative-is-not-riema – SmileyCraft Oct 17 '18 at 23:11
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Any such function can't have a jump discontinuity. But I am not sure if a function possesses an anti-derivative if it has discontinuities of oscillatory type. – Paramanand Singh Oct 18 '18 at 12:02