Extreme Value Theorem on a Compact Interval

Let $I$ be an interval in the sense of Interval in the Real Line, let $[a,b]\subseteq I$ with $a<b$, and let $f:I\to\mathbb{R}$ be continuous on $[a,b]$ in the sense of Continuity on a Closed Interval. Then there exist points $x_{\min},x_{\max}\in[a,b]$ such that