MA005: Calculus I

There are several reasons for us to examine continuous functions and their properties:

• Most of the applications in engineering, the sciences, and business are continuous and are modeled by continuous functions or by pieces of continuous functions.
• Continuous functions have a number of useful properties which are not necessarily true if the function is not continuous. If a result is true of all continuous functions and we have a continuous function, then the result is true for our function. This can save us from having to show, one by one, that each result is true for each particular function we use. Some of these properties are given in the rest of this section.
• Differential calculus has been called the study of continuous change, and many of the results of calculus are guaranteed to be true only for continuous functions. If you look ahead into Chapters 2 and 3, you will see that many of the theorems have the form "If $\mathrm{f}$ is continuous and (some additional hypothesis), then (some conclusion)".