Proofs that the Intermediate Value Theorem (IVF) implies the Least Upper Bound Property for an ordered field usually use a continuous function that is not uniformly continuous like here https://math.stackexchange.com/a/2388654/539499. Is IVF for uniformly continuous functions enough to prove the Least Upper Bound Property?
Let $F$ be an ordered field. $F_{>0}$ is the set of all positive elements of $F$. IVF for uniformly continuous functions is: "If $a,b,L \in F$, $a<b$, $f: [a,b] \to F$ is uniformly continuous and $f(a)<L<f(b)$, then there exists $c \in (a,b)$ such that $f(c)=L$"
A function $f: [a,b] \to F$ is uniformly continuous iff for all $\epsilon \in F_{>0}$, there exists $\delta\in F_{>0}$ such that for all $x,y \in [a,b]$, $|x-y| < \delta \implies |f(x)-f(y)|< \epsilon$.
We can't assume that a continuous function on $[a,b]$ is uniformly continuous as well because that that is not true for all ordered fields. It is false in $\mathbb{Q}$ for example.