Let $f$ and $f_n$, $n \ge 1$ be $\mathbb R \to \mathbb R$ functions. For any sequence $x_n$, $n \ge 1$ such that $\lim_{n \to \infty} x_n = x$ it follows that $$\lim_{n \to \infty} f_n(x_n) = f(x).$$ How to prove that $f$ must be continiuous? It's obvious, but I don't know how to prove it formally.
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you mean continuous "at $x$"? And it must be a typo in the second sentence – Sergey Zaitsev Jan 30 '23 at 08:25
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I want to check if $f$ is continious at $x$. I edited the question, I wrote limit incorrect – Maximax67 Jan 30 '23 at 08:31
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1also https://math.stackexchange.com/questions/117717/how-to-show-f-is-continuous-at-x-iff-for-any-sequence-x-n-in-x-converg – Sergey Zaitsev Jan 30 '23 at 08:34
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@geetha290krm Note that there is a sequence of functions in this question. – Gary Jan 30 '23 at 08:46