In this video, I explain two key theorems about real-valued functions on compact and connected domains: the Extreme Value Theorem (EVT) and the Intermediate Value Theorem (IVT). The EVT tells us that a continuous function on a compact domain attains both its infimum and supremum, meaning the function has a global minimum and maximum. This plays a crucial role in optimization, as it guarantees that continuous functions on compact domains will have absolute extrema.
For the proof of the EVT, we rely on earlier results about the compactness of the image of continuous functions. Since the function's image is compact, the image is closed and bounded, ensuring that the infimum and supremum both exist and belong to the set of outputs. (See th-cam.com/video/JKGHuum-EFo/w-d-xo.html)
Next, I present the IVT in a more general form than typically seen in Calculus. It tells us that a real-valued continuous function on any connected domain in a metric space will hit every value between any pair of outputs. The proof is quick and highlights that the image of a continuous function on a connected set must also remain connected, making it impossible to skip any value. (See th-cam.com/video/T-dC5bZ8Gjo/w-d-xo.html)
#mathematics #ExtremeValueTheorem #IntermediateValueTheorem #CompactSets #ConnectedSets #Continuity #RealAnalysis #Optimization #MathematicalTheorems #advancedcalculus