To prove:a continuous real-valued function (f) defined on set S has an image (S') that is an interval then S must be an interval.

Tools:definition of continuity, definition of a connected set, connected set in R is an interval,

If S' is not a singleton; if $\displaystyle f(a), f(b) \in S'$ such that $\displaystyle f(a) < f(b)$ and $\displaystyle f(c) \in S'$ such that $\displaystyle f(a) < f(c) < f(b)$, can it be shown that $\displaystyle a < c < b$?

I'm looking for something to get me started. Thanks.