9.1.2        The Intermediate Value Theorem
Continuity can be understood by connecting it to the Intermediate Value Theorem (IVT) and solving equations of the form $f(x) = 0$.
IVT: If $f$ is a continuous function on the interval $[a,b]$ and $f(a) \cdot f(b) \gt 0$ then there is a number $c \in (a,b)$ where $f(c) = 0$.
Mapping diagrams provide an alternative visualization for the IVT. They can also be used to visualize a proof of the result using the "bisection method."

Bisection and IVT vizualized with GEOGEBRA.
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