To say that $f:[1,2]\cup[3,4]\longrightarrow \Bbb R$ is continuous has a sense ?

Question

The question is probably obvious, but is there a sense to say that for exemple $$f:[1,2]\cup[3,4]\longrightarrow \Bbb R$$ is continuous on $[1,2]\cup [3,4]$ or not really ?

Answer 1

Yes this makes sense. Recall that a function $f$ is continuous on a set $S\subset \Bbb R$ if $f$ is continuous on every point $x_0\in S$ that's:

$$\forall\epsilon>0\;\exists \delta>0,\quad \forall x\in S:\; |x-x_0|<\delta\implies |f(x)-f(x_0)|<\epsilon$$

Answer 2

Yes, the same definition of continuity applies also to this set.

Note that $f:[1,2] \cup [3,4] \to \mathbb{R}$ is continuous if for every open set $U \in \mathbb{R}$ the set $f^{-1}(U)$ is an open subset of $[1,2] \cup [3,4]$.