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Let $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is continuous and injective. Prove (or disprove) that $f$ is a strictly monotone function.

I have tried to prove the monotonocity of $f$ for then by the injectivity of $f$ strict monotonicity will follow. However, I couldn't make any success.

I also tried to show that,

Let $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is not a strictly monotone function. Then show that $f$ is either discontinuous or not injective.

In this case also I couldn't proceed beyond some trivial observations.

Can anyone help me to prove it?


marked as duplicate by hardmath, Daniel Fischer Sep 14 '15 at 13:21

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  • 4
    $\begingroup$ Assume $f$ is continuous and not monotonic. Use the intermediate value theorem to show it is not injective. $\endgroup$ – Daniel Fischer Sep 14 '15 at 13:11
  • $\begingroup$ @DanielFischer: What are possible cases I need to consider if $f$ is not monotonic? $\endgroup$ – user 170039 Sep 14 '15 at 13:15

The second statement is logically equivalent to the first. I advise you to prove the first one, and to prove it by showing that if $f$ not monotonic and is continuous, it is not injective.

You start with the fact that $f$ is not monotonic, meaning that there exists some series of three points $x_0<x_1<x_2$ such that either $f(x_0) < f(x_1)$ and $f(x_2) < f(x_1)$ or $f(x_0) > f(x_1)$ and $f(x_2) > f(x_1)$.

From there and from the fact that $f$ is continuous, you can prove it is not injective.


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