Let $f \in \mathcal{C}(\mathbb{R})$ such that
$$\lim_{x \rightarrow -\infty} f(x) = +\infty \quad \text{and} \quad \lim_{x \rightarrow +\infty} f(x) = +\infty$$
a) Show that the set $\{x \in \mathbb{R} \mid f(x) \leq f(0)\}$ is closed and bounded.\\
b) Deduce that there exists $x_* \in \mathbb{R}$ such that $f(x_*) = \min\{f(x) \mid x \in \mathbb{R}\}$.