Methods for Determining Continuity

Deciding whether a function is continuous at a point admits three routes: verify the definition directly, use the limit test, or compare the left-hand and right-hand limits. They look like three ways of saying the same thing, but their applicable situations differ. The definition is the most fundamental, yet substituting into two limits often repeats the same computation; the limit test reduces the question to “does the limit exist and does it equal the function value”, which is the cheapest route once a limit is already known; the one-sided limit rule simplifies further — if both one-sided limits exist and are equal, the function is continuous there and no comparison with the function value is needed. This chapter also stresses a common misconception: being defined at a point and having a limit there does not imply continuity — the two must actually be equal. Deciding at the breakpoints of piecewise functions depends on exactly this point.