Tomorrow I’ll be introducing the intermediate value theorem (IVT) to my calculus class. Recall the statement of the IVT: if is a continuous function on the interval and is between and , then there exists a value such that . In other words, achieves all of the intermediate values between and .
This is a very underappreciated theorem by the students. They think that it is “obvious”—if a function is continuous, then clearly it will hit all of the intermediate values. That’s what continuous means, right?
I remind them that continuity is defined at a point—it is a local definition. I try to convince them that their intuition about continuity is global, and that it is precisely the IVT that supports their intuition.
As for applications of the IVT, I first remind them of the saying “the straw that broke the camel’s back.” That such a straw exists is an application of the IVT. I also have them prove that at every instant there are two antipodal points on earth that have the same temperature. (For each point on a given line of longitude define to be the temperature at minus the temperature at . Since is continuous and , there must be a location on this line of longitude at which . In other words, the temperature at and at are the same.)