Extreme Value Theorem

Definition

Definition in Order Topology

Consider a Continuous Function $f:(X,{\mathcal{T}}_X)\to (Y,{\mathcal{T}}_Y)$ between topological spaces where $Y$ is an ordered set in the Order Topology ${\mathcal{T}}_Y$. Then, if $X$ is compact, then $\exists c,d\in X\text{s.t.}\forall x\in X,f(c)\le f(x)\le f(d)$

Definition in Real Numbers

If $f$ is continuous on a $[a,b]$, then $\exists a_0,b_0\in [a,b],\forall x\in [a,b],f(a_0)\le f(x)\le f(b_0)$

If a real-valued function $f$ is continuous on the closed interval $[a,b]$, then $f$ must attain a Extremum, each at least once.