Sorry for the poorly-worded title.(adsbygoogle = window.adsbygoogle || []).push({});

I help tutor kids with pre-calculus, and they're working inverse functions now. They use the "horizontal line test" to see if a function will have an inverse or not by seeing visually if it's one-to-one.

I was thinking about what that might imply. If a function passes the horizontal line test (let's assume it's domain is ℝ and that it's continuous everywhere), then it must not have any extremum, since then it'd fail the test. So this would imply that:

[tex]f'(x) \le 0 \hspace{3pt}\mathrm{or }\hspace{3pt} f'(x) \ge 0 \hspace{3pt} \forall x[/tex]

I'm wondering if this is always true, and so if it's possible to prove that a bijective function must have a derivative that's either zero or positive/negative everywhere, and I'm wondering if the converse is also true.

I'm more of a physics-guy than a math-guy, but I do find math interesting and these are the types of questions that I like to think about. Any thoughts?

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# A continuous function having an inverse <=> conditions on a derivative?

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