The Intermediate Value Theorem

You start the day at 5°C and by noon it's 15°C. Was it ever exactly 10°C? Obviously yes — the temperature moved continuously from 5 to 15, so it had to pass through every value in between, including 10. That common-sense guarantee is the Intermediate Value Theorem, and it's one of the most useful facts in calculus. It proves that equations have roots, that a graph crosses a target line, that a continuous function can't skip over a value. Continuity is the magic ingredient: without it, a jump could leap right past 10. With it, every intermediate value is forced.