A continuous function $f:\Bbb R\to \Bbb R$ is injective if and only if it is strictly increasing or strictly decreasing

Asked Nov 08 '15 at 08:27

Active Nov 08 '15 at 08:48

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Is the following statement true?

A continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ is injective if and only if $f$ is strictly increasing or strictly decreasing.

If it's true give a proof. If it's false give a counter-example.

I think the answer is false but I just can't think of a counter-example. Any help would be useful.

edited Nov 08 '15 at 08:48

Jean-Claude Arbaut

23,277

asked Nov 08 '15 at 08:27

Pablo

2

Is $f$ assumed continuous? – Nov 08 '15 at 08:28
Sorry my bad. Yes function is continuous – Pablo Nov 08 '15 at 08:40
5

One direction should be almost obvious, the other should follow from the intermediate value theorem. See this question. – Jean-Claude Arbaut Nov 08 '15 at 08:44
Wait, so the statement is true? – Pablo Nov 08 '15 at 08:47
@Pablo yes, it is – Ben Grossmann Nov 08 '15 at 08:49
Thanks Everyone!!! I got a little bit confused because the answer in my book said it was false but didn't give a counter-example. Really appreciate the help!! – Pablo Nov 08 '15 at 08:53
What's the book?! – BrianO Nov 08 '15 at 08:57

A continuous function $f:\Bbb R\to \Bbb R$ is injective if and only if it is strictly increasing or strictly decreasing

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