While studying the properties of ordinal utility functions, I came across the following question.
Given a strictly increasing function $f : D \rightarrow \mathbb{R}$, where $D$ is an arbitrary non-empty subset of $ \mathbb{R} $, can one always find a strictly increasing function $g : \mathbb{R} \rightarrow \mathbb{R}$ that is defined everywhere on $\mathbb{R}$ and is equal to $f$ everywhere in the set $D$?
I feel that the answer should be positive, however, there might be some counterexample I'm unaware of.