Let $X, Y$ be topological spaces, $D \subseteq X$, and $f: D \to Y$ a continuous function. Suppose that $a \in X \setminus D$ is a limit point of $D$, and there is some $y \in Y$ such that $f(x) \to y$ as $x \to a$. Is it always the case that $\bar{f}: D \cup \{a\}\to Y$, $x \mapsto \begin{cases} f(x) & \text{if } x \in D \\ y & \text{if } x = a\end{cases}$ is continuous? Amann's analysis claims that it is always continuous, but I have trouble proving it if $X$ is not Hausdorff. For instance if $x \in D$, isn't it possible that any neighborhood $U$ of $x$ (in $D \cup \{a\}$)will contain $a$, so that $f(U)$ might not be in the desired neighborhood of $f(x). $
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