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?

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 $V$.

  • $\begingroup$ 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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