What purpose does the adjective "monotonically" serve in the context of a "monotonically increasing/decreasing function"?

Question

Why employ the adjective "monotonically" when referring to an "increasing" or "decreasing" function? Forsooth, if the function is indeed exhibiting an upward or downward trend, it is inherently "monotonic". Thus, it would seem that the term "monotonically" serves no purpose but to belabor the point, leading one to question its necessity.

Answer 1

I would use monotone increasing function instead of “monotonically increasing function.”

Monotone is an adjective, so it modifies the noun function. Same for increasing. Thus, a monotone increasing function is one that is increasing, and also monotone. It's redundant, yes, but unambiguous.

Monotonically is an adverb, so it modifies the adjective increasing. So a monotonically increasing function is one that is increasing in a monotonic fashion. That might be considered ambiguous.

I looked in Courant's calculus text and found the former, and Apostol's calculus text and found the latter. Stewart uses either “increasing,” “decreasing,” or “monotonic” to mean one or the other. Spivak doesn't use “monotonic” at all, just “increasing” or “decreasing.”

If the Wikipedia page bothers you, just edit it. I read the talk history, and it could use some work.

Answer 2

Reasonable question! A monotonic function literally is either an nonincreasing or a nondecreasing function.

If the function has been identified as nondecreasing or strictly increasing or increasing or the like, then the adjective "monotonic" (or the adverb "monotonically") is redundant, perhaps even ambiguous. Is the former suggesting that non-monotonic increasing functions exist? Is the latter (since the dictionary, non-mathematics definition of "monotonic" is "unchanging") suggesting that the second derivative is constant?

Similarly, a strictly monotonic function is a function that is either strictly increasing or strictly decreasing; here, appending "monotonic" or "monotonically" again doesn't add value.

@student91: a piecewise function that is increasing on each interval of definition but makes jumps downwards. I'd call such a function "increasing" but not "monotone"

This function is neither monotonic nor increasing, but is possibly injective.

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Addendum

Usage frequency sorted based on Google seach results:

1. "increasing function" -monotonically -monotone -monotonic

   1,270,000 results

2. "monotonically increasing function" -"monotone increasing function" -"monotonic increasing function"

   277,000 results

3. "monotone increasing function" -"monotonically increasing function" -"monotonic increasing function"

   78,000 results

4. "monotonic increasing function" -"monotonically increasing function" -"monotone increasing function"

   53,200 results

Answer 3

Monotonic function are function which preserve the ordering relation such that:

$$ x\leq y\Rightarrow f(x)\leq f(y)\text{ for increasing functions}$$

Or reverse the ordering relation:

$$x\leq y\Rightarrow f(x)\geq f(y)\text{ for decreasing functions}$$

For example, the function $\exp(-x) + \cos(x)$ is decreasing but not monotonically. The property of monotonic function it's use to manipulate inequality in all the domain of a given function. If you want to see for the derivative, is monotonic the function if the first derivative is either positive or negative for all the domain, the function $x + \sin(x)$ is positive because it's derivative is always equal or bigger than $0$.

Answer 4

It might be to avoid ambiguities for people in different disciplines. In order theory one says "order preserving" synonymous to "increasing" in analysis. But an "increasing" map is $f:P\to P$, where $P$ is a poset, such that $x\leqslant f(x)$ for all $x\in P$.

Answer 5

I guess that the wording monotonically increasing is influenced by a tradition which uses increasing in a strict sense, i.e., $x<y$ $\Rightarrow$ $f(x)<f(y)$. Instead of weakly increasing (or increaing in the wide sense) many people prefer to say non-decreasing. But then functions which are not increasing need not be non-increasing -- and in order to avoid such a confusion, people feel better with monotonically non-decreasing (alternatively, one could say nowhere decreasing although this does not seem to be common).

It seems to me that such nonsense is a relict from times where all orders were total orders, so that, e.g., non-negative as the logical contrary of $x<0$ is indeed equivalent to $x\ge 0$. Nowaday, in many branches of mathematics, partial orders are much more important than total orders.