If $g(x) \rightarrow L$ as $x \rightarrow y$, then $f(x) \rightarrow L$ as $x\rightarrow y$? Let $y \in A \subset B \subset \mathbb{R}$ and let $f: B \rightarrow \mathbb{R}, g: A \rightarrow \mathbb{R}$ be functions such that $g(x)=f(x)$ for all $x \in A$. If $g(x) \rightarrow L$ as $x \rightarrow y$, then $f(x) \rightarrow L$ as $x \rightarrow y$.

I need to either prove this or provide a counter-example. My gut says that if y is not a limit point, then this is false. But this material is fairly new and I am not understanding it well. Am I at least on the right track?
Consider $A=[0,\infty)$, $B=\Bbb R$, $g(x)=\begin{cases}1&amp;\text{if }x\ge0\\ 0&amp;\text{if }x&lt;0\end{cases}$, $f=\left.g\right\rvert_A$ and $y=0$.
Suppose that $B=\mathbb R$, $A=\{0\}$ and $g(x)=0$ for $x \in \mathbb R$.

Then as you suspected, $\{0\}$ is a limit point of $B$ and $\lim\limits_{x \to 0, x \in B} g(x)=0$. However the limit of $f$ at $0$ is not defined as $0$ is not a limit point of $A$.