Darboux's theorem

Darboux's theorem states that any continuous, smooth function, regardless of how complicated its shape, always has at least one point where its slope (or derivative) takes on every intermediate value between its minimum and maximum slopes. In simpler terms, if you imagine a curve that’s smooth and continuous, the slopes along it never jump unpredictably; instead, they change smoothly, covering all intermediate values. This means that even complex functions have predictable behaviors in terms of how steep or flat they are at certain points—ensuring a kind of consistency in their rate of change.