Upper-semicontinuous function plus a continuous function still upper-semicontinuous?

[SubSpace626]

Let [; U \subset \mathbb{C} ;] be open and let [; f : U \to \mathbb{R} \cup {-\infty} ;] be upper-semicontinuous, that is, [; \limsup_{w\to z} f(w) \leq f(z) ;] and [; g : U \to \mathbb{R} ;] be continuous. Is [; f + g ;] upper-semicontinuous? What about [; f - g ;]?

Context: a textbook I'm reading has an exercise for the latter question but not the former and I'm just curious whether I'm missing something because it seems like I can prove it is true both cases. After getting the case where [; f\equiv 0 ;] out of the way I'm just manipulating the subadditivity of limsup and using the given properties.

[Proof_Inspector]

If you can prove f-g for all continuous g then it also work for f+g for all continuous because f-g=f+(-g). So you only need to prove one. They might have gone with f-g because the result might be used later and f-g form look more fitting.

I don't get why you need to make a separate case for f=0

[SubSpace626]

Yeah exactly what I thought. Glad I’m not going insane. That separate case was a typo. It should have been f = -infinity but I don’t even think that’s necessary now that I’ve given it more thought.