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Let $X$ be a Hausdorff space. Suppose $f:X\rightarrow \mathbb{R}$ is such that $\{(x,f(x)):x\in X\}$ is a compact subset of $X\times \mathbb{R}$.

How would I show $f$ is continuous?

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up vote 6 down vote accepted

Show that the pre-image of a closed set under $f$ is closed (lift a closed set first to $X \times \mathbb{R}$ via the projection $X \times \mathbb{R} \to \mathbb{R}$, intersect the result with the graph and project down to $X$).

Alternatively, note that $p_X(x,f(x)) = x$ is a continuous bijection from the graph to $X$. The hypotheses imply that this is a homeomorphism. Then observe that $f = p_{\mathbb R} \circ (p_{X})^{-1}$ is a composition of continuous functions.

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