The study of functions falls into two general camps: the weird parts and the nice parts.

The nice parts fit together nicely. Expectations hold true. Predictions are easy. Models made with nice functions are easier to analyze and interpret.

The weird parts are where we learn just what the real numbers can do. The weird parts are difficult to predict. Analysis of weird parts means not trusting your intuition. Your descriptions must be deliberate and specific about what exactly is going on and what is not. Of course, mathematicians find the weird parts more interesting. It is even more interesting when a function is constructed only of weird parts.

For Precalculus, weird begins with discontinunities and singularities. What exactly is going on at a discontinuity or singularity?

To investigate discontinuities and singularities, we need to put our ideas of closeness into action.

Learning Outcomes

In this section, students will

• study removable discontinuities.
• study jump discontinuities.
• study asymptotic discontinuities.