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Let $f:(X,d)\to (Y,\rho)$.

Prove that $f$ is continuous if and only if $f$ is continuous restricted to all compact subsets of $(X,d)$. I could do the left to right implication but couldn't do the reverse implication. Please help.

Thanks in advance!!!

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If $x_n \to x$ in $X$, then $\{x_n\}\cup \{x\}$ is a compact subset of $X$, I guess... – Siminore Aug 29 '12 at 9:41
up vote 2 down vote accepted

Just as @Siminore pointed out in comments.

To be more precise:

assume that $f$ is continuous on every compact subset of $X$.

Suppose that a sequence $x_n$ of elements of $X$ is such that $x_n\rightarrow x$ for some $x\in X$. Observe that a set $\{x_n:n\in\mathbb{N}\}\cup\{x\}$ is a compact subset of the space $X$. Thus, along our assumption, $F$ is continuous on that set and it does simply means, that $f(x_n)\rightarrow f(x)$. Since the choice of a sequence was arbitrary we have proved that $f:X\rightarrow Y$ is a continuous function. Q.E.D.

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