Continuity of a constant function

Question

If f assumes only finite many values, then f is continuous at a point $x_0$ if and only if f is constant on some interval $(x_0 - \delta, x_0 + \delta)$

I know how to prove continuity for a given interval but I am having trouble with proving it is constant. Thanks for your help.

Answer 1

Suppose $f$ only takes on the values $c_1,\ldots,c_n$. Suppose that $f(x_0)=c_1$. Without loss, suppose that $|c_1-c_2|\leq |c_1-c_i|$ for all $i>1$. Choose $0<\epsilon<|c_1-c_2|$. Now run the definition of continuity.