There are two topology problems:
Let $X$ be a Hausdorff space. Let $f : X \to \mathbb{R}$ be such that $\{(x, f(x)) : x \in X\}$ is a compact subset of $X \times \mathbb{R}$. Show that $f$ is continuous.
Let $X$ be a compact Hausdorff space. Assume that the vector space of real-valued continuous functions on $X$ is finite dimensional. Show that $X$ is finite.
Please help, how can I solve these problems?