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**manjohn12** Suppose $\displaystyle I $ is an interval in $\displaystyle \mathbb{R} $ and let $\displaystyle f: I \to \mathbb{R} $ be a continuous function. Prove that $\displaystyle f $ is injective $\displaystyle \Longleftrightarrow f $ is strictly monotonic.

$\displaystyle \Rightarrow $ direction: Let $\displaystyle a,b \in I $. Suppose $\displaystyle a \neq b $. Then $\displaystyle f(a) \neq f(b) $. So $\displaystyle a<b $ or $\displaystyle a > b $. And $\displaystyle f(a) < f(b) $ or $\displaystyle f(a) > f(b) $. From here, would you use the intermediate value theorem?

$\displaystyle \Leftarrow $ direction: Suppose $\displaystyle f $ is strictly monotonic. Then $\displaystyle x \neq y \implies f(x) \neq f(y) $.