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If $f\in C^2[a,b]$ and $f(a)=f(b)=0$,then exist $\xi \in (a,b)$,such that $\int_{a}^{b}f(x)dx=-\frac{1}{12}f''(\xi)(b-a)^3$
It is the error formula of trapezoidal rule and can be easily proved by partial integration or error formula of Lagrange's interpolation.
What if we reduce the requirement of $f(x)$.$f(x)$ is twice differentiable(may not be continuous).The error formula still works but the two methods fail.How can I prove that then?

Mathscraft
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  • You mean that $f(x)^2$ is that smooth? How do you still get the same error formula if for instance the function switches sign in the middle The function could be piecewise linear with non-zero integral, while the second derivative is zero. – Lutz Lehmann Dec 07 '23 at 14:55
  • @LutzLehmann I think this is about $f$ being twice differentiable but $f''$ not necessarily continuous. – PierreCarre Dec 07 '23 at 16:15
  • Then this could be an unfortunate sentence structure and not the line dot replacing the multiplication sign. Then one can still construct counter examples with piecewise quadratic polynomials. – Lutz Lehmann Dec 07 '23 at 16:32
  • In the error formula, if $f''$ isn't continuous, the conclusion is still right.But in that situation, $f''$ may not be Riemann integrable. So the previous methods don't work. – Mathscraft Dec 07 '23 at 16:34
  • "the conclusion is still right." - how do you know that it is still right? Either you do not know it, in which case you should not be pretending that you do; or else you do know it, in which case you already have a proof available. Also, please recall that Riemann's is not the latest-and-greatest in definitions of the integral. For this, Henstock-Kurzweil would likely be the most useful. – Paul Sinclair Dec 08 '23 at 20:04
  • It is actually an exercise of my analysis course but I don't know the solution. Riemann's theory isn't the latest but I just wonder how can I prove it in Riemann Integral. – Mathscraft Dec 09 '23 at 02:08

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