How to prove the following problem:
Suppose $f \in PC(a,b)$, where $PC(a,b)$ means the set of piecewise continuous functions on the interval $[a,b]$ and $f(x) = \frac{1}{2}[f(x-) +f(x+)]$ for all $x \in (a,b)$. Show that if $f(x_0) \neq 0$ at some point $x_0 \in (a,b)$, then $f(x) \neq 0$ for all $x$ in some interval containing $x_0$. ($x_0$ may be an endpoint on the interval).
Best Answer
Hint: There is a closed interval $I$ containing $x_0$ on which $f$ is continous. Use continuity of $f$ in $I$ to prove that values near $x_0$ produce function values near $f(x_0)$, therefore away from $0$.