Functions such that $\lim_{n\to\infty}\frac{1}{n}(f(\frac{x}{n})+f(\frac{2x}{n})+...+f(\frac{nx}{n}))$=$f(x)$

Question

I need to find all $f(x)$ continuous function s that :

$\ \lim_{n\to\infty}$$\frac{1}{n}(f(\frac{x}{n})$+$f(\frac{2x}{n})$+...+$f(\frac{nx}{n}))$=$f(x)$

It looks similar to Riemann integral, but I'm not sure how to show that.

thanks.

Answer 1

For $x>0$ consider the interval $[0,x]. $. Then $\{\frac{jx}{n}: j=0,...,n\}$ is a partition of this interval, hence

$$\frac{x}{n}\sum_{k=1}^n f\bigg(\frac {kx}n\bigg) \to \int_0^x f(t) dt.$$

Therefore $$f(x)=\frac{1}{x}\int_0^x f(t) dt.$$

This shows that $f$ is differentiable for $x>0$ and (your turn !) $f'(x)=0$.

Hence, $f$ is constant .