Suppose that $f:[a,b]\to \mathbb{R}$ is continuous and
let $\displaystyle f_n(x)=\dfrac{x-a} {n}\sum_{k=1}^{n}f\left(a+\dfrac{k(x-a)}{n}\right)$
so, $f_n$ is uniformly convergence when $[a,b]$?
I think $f_n(x)$ is pointwise convergence, but I don't know that is uniform convergence or not..