Let $f: E \subseteq \mathbb{R} \to \mathbb{R}$ where $E$ is open. Let $x \in E$.
What is meant with $\lim_{h \to 0}f(x+h)$? More precisely, over what values can $h$ range?
I think there are two possibilities:
(1) Put $V:= \{h \in \mathbb{R} \mid x+h \in E\}$. Then $h$ can range over $V$.
(2) Because $x \in E$, and $E$ is open, there is $r>0$ with $(x- \epsilon, x + \epsilon) \subseteq E$, so $h$ can range over $(-\epsilon,\epsilon) \subseteq E$.
Am I correct that when one writes $\lim_{h \to 0} f(x+h)$, then one means either one of these two options?
It also seems that there is no ambiguity, because I managed to prove that
$$\lim_{h \to 0, h \in V} f(x+h) = \lim_{h \to 0, h \in (- \epsilon, \epsilon)}f(x+h)$$
Is all this correct?