Supremum of a continuous function on $[a,b]$

Question

Is the supremum of a continuous function $f$ on $[a,b]$ equal to the maximum of $f$ on $[a,b]$? What about a bounded function on $[a,b]$?

Answer 1

Be because $[a,b]$ is compact, and $f$ continuous, $f([a,b])$ is compact. In $\Bbb R$ compact is equivalent to closed and bounded (above and below). So, $f$ will take on a minimum and maximum value. $\sup f([a,b])$ is a limit point of the set, and so is in the set since it is closed. Therefore, the sup is the max in this situation.