An increasing function is ALWAYS continuous at SOME point!
We rigorously prove that an increasing (or decreasing) real-valued function of one real variable is continuous SOMEWHERE. In fact, we prove that it is continuous ALMOST EVERYWHERE (the set of points of continuity is very large)! The intuition is that the discontinuities of such a function should be jump discontinuities and there is not enough space for the graph of the function to jump too many times. We make this intuition rigorous and execute the mathematical proof in two steps.
The video is designed for those interested in theoretical calculus, real analysis, and proof-based math, and is designed to be as accessible as possible to demonstrate both intuition and rigor in mathematical proofs.
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