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 $x_0$.