connected set of sum of upper semi continuous function

Question

Let $C(X)$:space of continuous functions on a compact space.

Consider $f$ and $g :C(X)\rightarrow \mathbb{R}$ are upper semi continuous.

suppose for every $T\in C(X)$ set of $f(T)$ and set $(f+g)(T)$ are closed interval(connected set).Can we say set of $g(T)$ is closed interval(connected set),as well?

If Not,under which condition we have it.

Answer 1

The answer is no. First ask this question for functions from $\mathbb{R}$ to $\mathbb{R}$. There are easy counterexamples: take $f(t) = t$ and let $g$ be the characteristic function of the interval $(-\infty,0)$. Then the ranges of $f$ and $f + g$ are both $\mathbb{R}$ but the range of $g$ is not connected.

You can turn that into a counterexample on $C(X)$ by fixing a point $x \in X$ and composing the $f$ and $g$ described above with the evaluation functional $\phi \mapsto \phi(x)$ from $C(X)$ to $\mathbb{R}$ (or, really, any nonzero linear functional on $C(X)$).

I can't think of any good condition that would guarantee this. Just knowing that the range of a function is connected tells you very little about the function.