I'm wondering if someone can help me prove that if f is strictly monotone, then f is injective?
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1Suppose $f$ is not injective. That means there must exist distinct $a,b$ such that _____? – quasi Apr 06 '17 at 21:43
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Do you know what are the definitions of monotone and injective? – Crostul Apr 06 '17 at 21:51
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Beware of the context... Giving an example of a strictly monotone function which is not injective. – Anne Bauval Sep 21 '23 at 16:34
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Well, is it possible to have $f(a) < f(b)$ and also $f(a) = f(b)$?
strictly monotonically increasing [decreasing] means when ever $a < b$ then $f(a) < f(b)$ [$f(a) > f(b)$].
So if $a \ne b$ then either $a < b$ or $a > b$ and if $f$ is monotonic, either increasing or decreasing, then eithere $f(a) > f(b)$ or $f(a) < f(b)$. In either case $f(a) \ne f(b)$.
fleablood
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