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## Homework Statement

Let ##a,b\in \mathbb{R}##, ##a<b## and let ##f## be a continuous real valued function on ##[a,b]##. Prove that if ##f## is one-one then ##f([a,b])## is either ##[f(a),f(b)]## or ##[f(b),f(a)].##

**2. The attempt at a solution**

Suppose ##f(a) < f(b)## then by IVT we have if ##x\in(a,b)## then ##f(x) \in [f(a),f(b)]##. Arguing by contradiction suppose ##f(x) < f(a)## or ##f(x)>f(b)##. If ##f(x) <f(a)## then by IVT ##f([x,b])## contains all points between ##f(x)## and ##f(b)## and contains ##f(a)## contradiction that ##f## is one to one.

Why does that contradict the assumption that ##f## is one to one?