$f:R\to R$ is a continuous function such that for every $r>0$ and $x\in R$.
If $\frac{1}{2r}∫^{x+r}_{x-r}f(t)dt=f(x)$, show that there exist constants $a,b$ such that $f(x)=ax+b$.
I tried to do this by fixing x at $x_0$ and r at $r_0$ and tried the contrapositive approach but got stuck. Can someone guide me in the right direction...as in how to start on this problem...and what ideas would be relevant in trying to prove this.
I understand what is happening. The mean value of the function is basically the value of the fucntion at the center of the interval. I just don't know how to start on the proof.