(At the outset of this question, I will note that my background in analysis is introductory, derived from the first 11 chapters of Spivak's Calculus.)
I recently proved the following theorem:
If the function $f: \mathbb R \to \mathbb R$ has the property that for any $x \in \mathbb R$ there is a $\delta_x \gt 0$ such that on the interval $(x-\delta_x,x+\delta_x)$ $f$ is strictly increasing, then $f$ is a strictly increasing function $(\dagger)$.
Because Spivak has not introduced topological notions of "covers" or "compactness" (which seems to be the conventional way of proving this statement), I relied on the Least Upper Bound Property (LUBP), as Spivak covers this in Chapter 8.
This was a proof by contradiction, supposing there was some $x \lt y$ such that $f(x) \geq f(y)$. Ultimately, I used the following set to produce the contradiction:
$$S=\left\{z \in \mathbb R \ | \ z \gt x \text{ and } \forall w \in (x,z]\Big (f(x) \lt f(w) \Big)\right\}$$
After finishing this proof, I wondered whether or not the Least Upper Bound Property was the only reason I could carry out this proof.
I considered modifying $(\dagger)$ to $f: \mathbb N \to \mathbb N$ and $f:\mathbb Z \to \mathbb Z$. With some slight changes (e.g. requiring $\delta_x \gt 1$), it seems like the claim still remained valid despite not being able to invoke the LUBP (I suspect this is related to intervals in $\mathbb N$ and $\mathbb Z$ having some sense of finiteness...i.e. I can count the number of elements between $x$ and $y$ in the interval $[x,y]$).
I then came to modifying $(\dagger)$ to $f: \mathbb Q \to \mathbb Q$. To prove $(\dagger)$ in the original setting of $\mathbb R$, the LUBP seems to have let me sneak "right on top of" the set $S$, from a starting point of $y$. However, in the setting of $\mathbb Q$, where $S$ would be redefined as:
$$S_{\mathbb Q}=\left\{z \in \mathbb Q \ | \ z \gt x \text{ and } \forall w \in (x,z]\Big (f(x) \lt f(w) \Big)\right\}$$
(where $(x,z]$ should be understood to only contain elements of $\mathbb Q$) it does not seem to me that I will be able to get right on top of $S_{\mathbb Q}$.
Should I conclude that the claim is false for $\mathbb Q$? If so, might I have a counter example? Further, if false, then, given that the claim appears to be true in $\mathbb N$ and $\mathbb Z$, should I conclude that if my setting involves intervals $[x,y]$ that contain an infinite number of elements but lacks some flavor of the LUBP, the claim will always be false?