For every sequence $\{x_n\}$ in $S$ with $\lim x_n =c$, $\{f(x_n)\}$ converges. Show that $f$ is continuous at $c$.

Question

Let $f:S \to \mathbb{R}$ be a function and $c \in S$. For every sequence $\{x_n\}$ in $S$ with $\lim x_n =c$, $\{f(x_n)\}$ converges. Show that $f$ is continuous at $c$. My question is for what definition of $S$ would this statement be valid? For example, if $S \subset \mathbb{R}$, then define a function $f(x)=1$ if $x \neq 0$ and $f(0)=0$ which would satisfy the convergence condition but not be continuous at $c=0$. Or am I missing something?

Answer 1

It completely depends on whether you're allowed to take $x_n = c$ or not. If the sequence is not allowed to take the value $c$, then I'd agree that your function is a counterexample. If not, then take the sequence $$(1, 0, 1/2, 0, 1/3, 0, 1/4, 0, 1/5, \ldots)$$ which maps to the sequence $$(1, 0, 1, 0, 1, 0, \ldots)$$ under $f$, which is not convergent! If the question allows for $x_n = c$ (and it doesn't appear to forbid it), the result is true.