Laying the Foundation for Calculus
I have some good news and some bad news. The good news is that we’ll be dealing with relatively easy functions for the remainder of calculus. The bad news is that we need to mathematically define exactly what we mean by “easy.” When math people sit down to define things, you know that theorems are going to start flying around, and calculus is no different.
When we say “easy” functions, we really mean continuous ones. In order to be continuous, the function can’t contain holes and isn’t allowed to have any breaks in it. That sounds nice, but math people like their definitions more specific (read: complicated) than that. In order to define “continuous,” we’ll first need to design something called a “limit.” During this part of the book, you’ll learn what a limit is, how to evaluate limits for functions, and how to apply limits to design a definition for continuity.