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The Newton-Leibniz formula requires from a function $f\colon\left[a,b\right]\to\mathbb{R}$ to be integrable (Riemann-Integrable) and to have an antiderivative $F$ over the interval $\left[a,b\right]$. Then we get: $$ \int_{a}^{b}f(x)dx=F(b)-F(a) $$ I was wondering,

  1. What kind of integrable functions don't have an antiderivative?
  2. What kind of non-integrable functions have an antiderivative?
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For the first point note that derivatives don't have simple discontinuity so we just need to create discontinuous functions which are integrable and have simple discontinuity. Thus for example consider $f(x) =\lfloor x\rfloor $ on $[0,2]$.

The second point requires much more work and it is not easy to find such a function by trial and error. Historically no one believed that such a function existed until Vito Volterra created one such function. The Volterra function is differentiable everywhere with a bounded derivative and the derivative is not Riemann integrable on any closed interval. So what you need is the derivative $V'$ of the Volterra function $V$.

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  • $\begingroup$ Modulo a set of measure $0$, which anyway doesn't affect integrals, $f$ does have an antiderivative $F(x) = \int_0^x f(t) \, dt.$ The derivative of $F$ only differs from $f$ in two points: $1$ (where it's not defined) and $2$. $\endgroup$ – md2perpe Dec 1 '17 at 16:16

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