If $f,g : \mathbb [0,1] \to \mathbb R$ are lower-semicontinuous, and $f(x) = g(x)$ for all $x$ in a dense subset of $[0,1]$, is it true that $f(x) = g(x)$ everywhere in $[0,1]$? If it helps, we can also assume that $f(x) \leq g(x)$ everywhere.
The context is this question, where $M$ is the pointwise supremum of a family $\mathcal F$ of continuous functions. I have constructed an increasing sequence $(f_n)$ of functions in $\mathcal F$ such that $f_n \to L$, and $L(q) = M(q)$ for each $q \in [0,1] \cap \mathbb Q$. So, $L$ and $M$ are LSC and agree on a dense subset, and $L \leq M$ pointwise, but I don't know if this implies agreement everywhere.
I've attempted to find a proof but haven't been able to make it work and I'm not sure whether it's actually true. I would appreciate either a hint or, if it's false, a counterexample and perhaps an additional hypothesis (less restrictive than continuity) that would make it true.
