What does the intermediate value theorem guarantee about a continuous function?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

In analysis, the intermediate value theorem guarantees that if a real-valued function is continuous on a closed interval, it must take on every value between the values of the function at the endpoints of that interval.1

Less formally, this means that an unbroken curve cannot move from one height to another without passing through every intermediate height along the way. For example, if a continuous function $f$ defined on an interval $[a, b]$ satisfies $f (a) < 0$ and $f (b) > 0$ , the theorem guarantees there exists at least one number $c$ between $a$ and $b$ such that $f (c) = 0$ .1