Existence of such points in compact and connected  topological space $X$ Let $X$ be a topological space which is compact and connected.
$f$ is a continuous function such that;
$f : X \to \mathbb{C}-\{0\}$.
Explain why there exists two points $x_0$ and $x_1$ in $X$ such that $|f(x_0)| \le |f(x)| \le |f(x_1)|$ for all $x$ in $X$.

Define the function $g: X \to \mathbb{R} $ by $g(x) = |f(x)|$, which is continuous.  Since X is compact, the result follows by the Extreme Value Theorem.
