We begin with a close look at the real number system. We look at the subset structure of the real number line. Some sets have interior points'', which are surrounded only by other points in the same set. Some sets have no interior points. We define continuous functions as those which don't rip apart'' interiors of sets.
Analysis also involves approximating functions by sequences and series of other functions.
Another important topic in analysis involves the fact that there are contexts in which some intervals of real numbers are more important than other intervals of the same length. We learn how to measure'' subsets of the reals with different kinds of measures, and then learn to integrate with respect to different measures. It's possible to obtain several different values when you integrate a single function, depending on the measure used in the integration process. These ideas are at the foundation of probability theory and spectral analysis. When a chemist or biologist uses an electron microscope to examine a piece of material that is utterly ordinary, she can discover an entire world of complexity and beauty beyond her imagination.