[Math] Can a function be increasing or decreasing at a point

Question

I was solving:
Determine the intervals of increase and decrease for $f(x) = \frac {2x}{ ln x}$ and I stumbled upon the fact that f(x) is decreasing on (0,e] and increasing on [e, $\infty$). This would otherwise suggest that the function is both increasing and decreasing at x=e. Is that true? Or a function can be sure to be increasing or decreasing only in the vicinity of a point but at a point?

Answer 1

A function can't be increasing or decreasing unless you can compare it to another point.

So it depends on definition.

I believe there are 3 (or 4... or 5...) incompatible options.

1) Increasing at a point $x$ means that there is an $\epsilon > 0$ so that for every $x-\epsilon < y < x < z < x+\epsilon$ such that $f(y) \le f(x) \le f(z)$. (strictly increasing would mean strict inequalities.) and Increasing on an an interval would mean increasing on every point of an interval.neither at $e$.

However I have NEVER seen anyone or any text use this definition.

In fact, I just made it up.

2) Increasing at a point is a logical inconsistancy and makes no sense. Increasing on an interval (whether open or closed or mixed) means for any two points $x,y$ in the interval so that $x<y$ then it must follow that $f(x) \le f(y)$.

This seems to be the most accepted definition. So $f$ is decreasing on $(0,e]$ and decreasing on $[e,\infty)$.

And increasing at $x = e$ simply is not a meaningful concept.

2a) Same as above but allowing "increasing at a point" to mean the point is within and interval where the function is increasing.

In this case $f$ is both increasing and decreasing "at" $e$. I've seen people say this but it's really semantics and not mathematics.

2b) Same as above but allowing $\{e\}$ to be a "single point interval". THus every function is vacuuously both increasing and decreasing at every point because there are no $x < y$ in the "interval" than for all (all zeor of them) $x < y$ we have $f(x) \le f(y)$. (We also have $f(x)$ is a blue dragon eating colorless yellow thoughts.)

Again... semantics; not mathematics.

3) $f$ is increasing on a set of points $S$ so that for any $x,y \in S$ and $x< y$ then $f(x) < f(y)$.

This would mean vacuously that every function is increasing and decreasing on a set with a single point.

However I have never seen anyone use this definition and I just made this up. It is probably useless as I can say something line $f(x) = x$ if $x \in \mathbb Q$ and $f(x) = 0$ if $x$ is irrational, would be increasing on the rationals. WHich I think avoids the issues.

Any way...

I think most would use definition 2. (But there are always exceptions.) But practically, I don't think claiming $f$ is increasing at a single point makes much sense or is useful unless you are claiming the point is in an interval on which the function is increasing.