Weierstrass Extreme Value Theorem

The Weierstrass Extreme Value Theorem states that if a function is continuous on a closed and bounded interval, then it must reach both its highest and lowest points within that interval. In other words, there are specific points where the function attains its maximum and minimum values, without just approaching them but actually taking those exact values somewhere in the interval. This guarantees predictable, well-defined peaks and valleys for continuous functions over finite, closed ranges.