Intuition for Theorems About Continuous Functions - Expii
There are at least a couple different ways to gain intuition for continuous functions. One is by drawing or visualizing a graph y=f(x). Another is: Suppose you're drawing back and forth on a number line (a one-dimensional figure) without ever lifting your pencil. Then your pencil's position on the number line is continuous as a function of time. With this intuition in mind, facts such as the intermediate value theorem (IVT) are much easier to digest.