Can we say that a function is increasing/decreasing at a point where tangent is vertical? I am analyzing this function: $f(x)=\sqrt[3]{x^3-x}$. The derivative is $f'(x)=\frac{3x^2-1}{3\sqrt[3]{(x^3-x)^2}}$. I found that $f'(x)>0$ when $x<-1 \text{ or } -1<x<-\frac{1}{\sqrt 3}$. Can I say that the function is increasing on interval $(-\infty, -\frac{1}{\sqrt 3})$ (include $x=-1$ even though the derivative does not exist at that point? I am thinking that the function continues to increase. It's been awhile since I did any analysis like this but I believe the definition of increasing function does not involve the derivative. Am I right?
 A: I think the disconnect here is that you're thinking about a specific point. Usually, when we talk about increasing/decreasing functions, or monotonicity more generally, we're typically referring to an entire interval and looking at pairwise comparisons on that interval - not just at one point.
Further, the existence of the derivative is not a requisite condition for a function to be considered increasing.
One common definition of increasing is that a function $f$ is increasing on an interval $I$ if for any $x,y \in I$,  $y>x \Rightarrow f(y) \ge f(x)$, and $f$ is strictly increasing if $y>x \Rightarrow f(y) > f(x)$.
Note, this definition has nothing to do with derivatives, and the definition applies to a function over some interval - not just a single point!
The floor and ceiling functions are (weakly) increasing over $\mathbb{R}$, but their derivatives are either $0$ or undefined for all $x\in \mathbb{R}$.
As KCd noted in their comment, we can use derivatives to arrive at a sufficient condition. If $f$ has a derivative that exists on an interval, and $f'(x) \ge 0$ everywhere on that interval, then we can show that $f$ is increasing between any two points on that interval based on the definition above, but the converse does not need to be true - a function can be increasing even if the derivative doesn't exist or is undefined.
Using the sufficient condition $f'(x) \ge 0$ makes it easy to think about monotonicity at a point because we typically evaluate derivatives at a point, but it really is a property of a function on an ordered set or an interval, not at a specific point.
