Consider $f: [a, b] \to \mathbb{R}$ as a function that remains continuous over the closed and bounded interval $[a, b]$.
Demonstrate that the range of values of $f$, denoted $f([a, b])$, possesses an upper bound. In simpler terms, prove the existence of a real number $M$ such that for every $x$ within the interval $[a, b]$, $f(x)$ is less than or equal to $M$.
