Tuesday, 13 September 2016

If a function $f: A \subset\mathbb{R}\rightarrow\mathbb{R}$ is strictly increasing, then $f$ is injective.

Theorem: Consider a function $f: A \subset\mathbb{R}\rightarrow\mathbb{R}$ that is strictly increasing. Then $f$ is an injective function.

Proof: We know that $f$ is strictly increasing on its domain $A$. Thus, for $x_1 < x_2$, we have $f(x_1) < f(x_2)$. So it's clear that if $x_1 \neq x_2$, then $f(x_1) \neq f(x_2)$. Thus $f$ is injective.

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