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To clarify, I know what it means when a function is surjective, injective, bijective, and what its inverse is. This is something my module notes covers.

However, I have a question (past exam paper) asking to show that a function is monotone. I have no clue what it means, and there is not a single mention of it in the module notes (provided by the professor, and I don't have access to a recommended textbook either).

Please could someone explain what monotone means?

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A real valued function $f$ of a real variable is monotonically increasing if $$ f(a) \ge f(b) \text{ when } a > b. $$ "Monotone" might be monotonically increasing or monotonically decreasing. Sometimes you want the inequality to exclude equality. Then you use the adjective "strictly".

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Perhaps that you are talking about monotonic functions. These are functions from an ordered set into an ordered set which either preserve the order or invert it. For instance,$$\begin{array}{rccc}f\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&\pm x^n\end{array}$$is monotonic if and only f $n$ is odd.

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A monotonic function is one that preserves some sort of well-defined "order". Some examples are:

  • a monotonically increasing function $f$, such that $f(x)<f(x+\epsilon)$ for all $\epsilon>0$,
  • a monotonically decreasing function $f$, such that $f(x)>f(x+\epsilon)$ for all $\epsilon>0$.
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  • $\begingroup$ What about $f(x)=x$, if x rational and $f(x)=\tan(x) $ otherwise in $x=0$? It's true that for any positive $\epsilon$ we have $f(x)<f(x+\epsilon)$, but $f$ isn't monotonic at all. $\endgroup$ Aug 22, 2019 at 15:40

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