# connected set of sum of upper semi continuous function

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.

• What does "set of $f(T)$" mean? If $f$ and $T$ are both fixed then $f(T)$ is a number. Sep 22, 2018 at 13:28
• @NikWeaver Yes f(T) is number.I mean $\{ f(T): T\in C(X) \}$is closed interval.Is it clear?
Sep 22, 2018 at 16:45
• Got it, thank you. Sep 22, 2018 at 17:25
• You're welcome,
Sep 22, 2018 at 17:56

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.

• If $f = g$ and you assume the range of $f$ is connected, then the range of $g$ will be connected. What is the question? Sep 22, 2018 at 19:01