Arithmetic preserves continuity

Let \( X \) be a subset of \( \mathbb{R} \), let \( f : X \to \mathbb{R} \) and \( g : X \to \mathbb{R} \) be functions, and let \( x_0 \) be an element of \( X \). Then, if \( f \) and \( g \) are continuous at \( x_0 \), then so too are the functions [...], [...], [...], [...] and [...]. If \( g \) is non-zero on \( X \), then [...] is also continuous at \( x_0 \).