# What does it mean by 'monotone' when referring to functions?

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?

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".
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.
• a monotonically increasing function $$f$$, such that $$f(x) for all $$\epsilon>0$$,
• a monotonically decreasing function $$f$$, such that $$f(x)>f(x+\epsilon)$$ for all $$\epsilon>0$$.
• 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. Aug 22, 2019 at 15:40