Continuous Real-Valued Functions on a Compact Interval are Bounded

Let $a$ and $b$ be real numbers with $a<b$, and let $g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric. Then there is a real number $C\ge0$ such that $|g(t)|\le C$ for all $t\in[a,b]$.