If $f:[a,b]\to \mathbb{R}$ is continuous on $[a,b]$, show that $\mathrm{\exists}c\in [a,b]$ so that

${\int}_{c-a}^{b-c}f(x)\phantom{\rule{thinmathspace}{0ex}}dx=0$

I think it can be solved using the intermediate value theorem but I can't find a suitable function.

${\int}_{c-a}^{b-c}f(x)\phantom{\rule{thinmathspace}{0ex}}dx=0$

I think it can be solved using the intermediate value theorem but I can't find a suitable function.