Topic: Continuous and uniformly continuous functions between metric spaces
(a) If $f: (X, d_1) \to (Y, d_2)$ is an isometry, it proves that $f$ is an immersion
and if $f$ is surjective, then it is a homeomorphism.
(b) Suppose that $f: \mathbb{R} \to \mathbb{R}$ is a function, and consider the function $|f|: \mathbb{R} \to$
$\mathbb{R}$ given by $|f|(x) = |f(x)|$. If $|f|$ is continuous at a point $x_0$, so is it valid
that $f$ is continuous at $x_0$? Is the reverse true?
