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In BBFSK, (~1960, Germany) at least in the section I am currently reading, the authors use the term monotone increasing (decreasing) to mean what I often see called strictly increasing (decreasing). I'm not sure where I first learned to use the term monotone in the mathematical sense, but I believe it was first introduced to me as having the same meaning as strictly monotone. That is, a monotone function is nowhere constant. In contrast a function which is everywhere increasing or constant is called non-decreasing.

Checking an edition of Anton's very respectable Calculus book shows he uses the term to mean non-increasing(decreasing). For me, that renders the term monotone virtually useless.

So I am wondering how much consistency there is in the field of mathematics regarding the definition of monotone. Has this changed over the years, or does it vary from one language to another?

After posting, I checked my own list of definitions and found that I have it recorded that monotone means that the derivative never changes sign. Strictly monotone means that the derivative is either everywhere positive, or everywhere negative. Monotone increasing (decreasing) means the derivative is everywhere positive (negative). So monotone increasing (decreasing) means strictly monotone.

Since I wrote that over a decade ago, I don't recall what my source was for that.

I also not that monotone often occurs in discussions of functions which do not have derivatives. Such as natural number addition.

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  • $\begingroup$ More appropriate for this cite. $\endgroup$ – tarit goswami May 17 at 17:25
  • $\begingroup$ Re your records: Note that strictly monotone does not even imply that the derivative exists. Also, $x\mapsto x^3$ is strictly monotone, but the derivative has a zero. $\endgroup$ – Hagen von Eitzen May 17 at 17:37
  • $\begingroup$ @HagenvonEitzen Indeed. I forgot to mention that. I have now added a statement to my question acknowledging that monotone is also used to describe functions which do not have derivatives. $\endgroup$ – Steven Thomas Hatton May 17 at 17:43
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From a very quick research that I did, I found that most people use monotonically increasing for what you would call non-decreasing (and vice-versa).

See for instance

However, it might be worth to explicitly mention if one is referring to the strict or non-strict variant since there seem to be also some texts that use the term increasing for strictly increasing.

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  • $\begingroup$ Ha! See my updated question. Apparently my notes agree with the last form you mentioned. $\endgroup$ – Steven Thomas Hatton May 17 at 17:37
  • $\begingroup$ I don't mind providing my own definition of how a term is to be understood in context, if there is inconsistency between authors. I prefer not to redefine a term that has a widely accepted meaning. The phrase strictly monotone increasing has a superfluous adjective. $\endgroup$ – Steven Thomas Hatton May 17 at 18:11

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