Let's say $f:[-1,1]\to\mathbb R$ is a bounded function. Let's say that $f$ is Riemann integrable over $[a,b]$ for every $a$ in $]-1,0[$ and every $b$ in $]0,1[$. Can we conclude that $f$ is Riemann integrable over $[-1,1]$?
I was first looking for a counter example. I see that the critical points are $-1,$ $1,$ and $0.$ I was looking for something with more than one discontinuity point because then it's not Riemann integrable.
I don't really see a counter example can someone maybe give me a hint.
Edit: Because of the information I get in the reaction I'm no longer looking for a counter example but I'm trying to prove that $f$ is Riemann integrable over [-1,1]. Now I tried to prove that the lower sum equals the upper sum, because that's a basic definition for Riemann integrability but I'm a little bit confused because we don't have a concrete function. Maybe someone can suggest another way for a proof?