I tried to solve this problem but I could not continue
Let $f: [a, b]\to \mathbb{R}$ be a continuous function. Prove that if $x_1, x_2,\ldots,x_n \in (a,b)$, then exists $x_0 \in (a, b)$ such that $f (x_0) =\frac{f (x_1) + f (x_2) +\cdots + f (x_n)}{n}$
the function is continuous, so $\forall\ x_1, x_2,\ldots,x_n \in(a, b)$ $\begin{align*} \lim_{x \to \ x_1}f(x)&=f (x_1)=L_1\\ \lim_{x \to \ x_2}f(x)&=f (x_2)=L_2\\ \lim_{x \to \ x_3}f(x)&=f (x_3)=L_3\\ &\vdots\\ \lim_{x \to \ x_n}f(x)&=f (x_n)=L_n\\ \end{align*}$
We have that:
$\lim_{x \to \ x_0}f(x)=f (x_0)=\frac{f (x_1) + f (x_2) +\cdots + f (x_n)}{n}= \frac{1}{n}\sum_{j=1}^n f(x_j)$ Now $\min\left \{ L_1,L_2,L_3,\cdots,L_n \right \}\le\frac{1}{n}\sum_{j=1}^n f(x_j)\le\max\left \{ L_1,L_2,L_3,\cdots,L_n \right \}$
and at this point I'm lost ...