continuity in an interval? Let $g : [a, b] → \mathbb{R}$ be a continuous function on $[a, b]$. Given any $n \in \mathbb{N}$ and
$x_1, . . . , x_n ∈ [a, b]$, show that there exists $x_0 ∈ [a, b]$ such that
$g(x_0) =(g(x_1) + · · · + g(x_n))$.
I am very confused on how to start here. I know that there is the basic epsilon delta proofs for continuity, but I don't how how or even if that should be applied here. 
 A: Either the numbers $g(x_1),\ldots,g(x_n)$ are all the same or at least two of them differ.  If they're all the same, then the average also has that same value, so you can take $x_0$ to be equal to $x_1$ or to $x_2$, etc.  But if they're not all the same, then there is a smallest one and there is a largest one.  The average is somewhere between those.  So you can use the intermediate value theorem to infer the existence of the desired point $x_0$.
If $g(x_i)$ is the largest among $g(x_1),\ldots,g(x_n)$ and $g(x_j)$ is the smallest, then the interval whose endpoints are $x_i$ and $x_j$ is the one to which you need to apply the intermediate value theorem.  Suppose $A=(g(x_1)+\cdots+g(x_n))/n$.  Then $A$ is between $g(x_i)$ and $g(x_j)$, so the intermediate value theorem says there is some number $x_0$ between $x_i$ and $x_j$ such that $g(x_0)=A$.
A: Since $g$ is continuous, there exists two real numbers $A,B$ such that $A\le g(x)\le B$ for all $x\in [a,b]$. It follows that
$$
\frac1n\sum_{i=1}^nA\le\frac1n\sum_{i=1}^ng(x_i)\le\frac1n\sum_{i=1}^nB,
$$
i.e.
$$
A\le \frac1n\sum_{i=1}^ng(x_i)\le B.
$$
Hence, thanks to the Intermediate Value Theorem, there is some $x_0\in [a,b]$ such that
$$
g(x_0)=\frac1n\sum_{i=1}^ng(x_i).
$$
