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 ...