IVT: If is continuous, and t is between f(a) and f(b), then there exists c in (a, b) such that f(c) = t.
What it means is that a continuous function can't "miss" any values.
Ex: Imagine this conversation:
Farmer: "I'm angry at you, Mr Weather Service spokesman. The temperature dropped from 60 degrees to 29 degrees in one day. I had an app set up with the Weather Service to text me if the temperature ever hit 45 degrees, so that I could prepare to defend my crops from the unusal cold. But I never got any text message."
Weather Service spokesman: "Don't blame us. The temperature never hit 45 degrees."
Farmer: "Nonsense. The temperature dropped from 60 to 29, so it MUST have been 45 SOMETIME during that drop. That's because temperature changes CONTINUOUSLY. It didn't skip or leap over any values during its drop from 60 to 29!"