> 
> Let $f$ be a function such that $|f(u)-f(v)| \leq |u-v|$ for all $u$ and $v$ in the interval $[a,b]$.
> 
> 
> Prove that
> $$\left|\int\_{a}^{b} f(x) dx- (b-a)f(c)\right| \leq\frac{(b-a)^2}{2}$$
> where $c\in [a,b]$.
> 
> 
> 

Is f continuous?

Any hint or suggestion will be appreciated.