Today I continued to work through the exercises in the appendix of my calculus book that provides a review of some concepts from algebra.
One series of problems involved finding the intervals on which various expressions containing variables would be less than or greater than zero. One method to do this that was outlined in the textbook was to find the points at which the expression would be equal to zero, use those to divide the number line into intervals, then test a point from each interval to determine whether the expression was positive or negative on that interval. All the time I was doing this I was thinking, “this is all well and good for expressions that correspond to continuous functions, but what about other types?” (The property of continuous functions that makes this work is the Intermediate Value Theorem, I believe.)
Sure enough, near the end of the series of problems, some expressions appeared that would be discontinuous when interpreted as functions. The answer to my question is that you need to treat each point of discontinuity as a point at which to divide the number line, the same as each point where the expression would equal zero. This was never explained or even hinted at in the text, though, which puzzles me. I don’t think of this text as leaving much for the reader to discover. Maybe there is more than I remembered, though. If so, there may be value in doing all of the exercises.
