2 LIMITS

2.5 CONTINUITYSUGGESTED TIME AND EMPHASIS1–1 1 2 classes (1 class for an overview, 1 1 2for a more detailed treatment)POINTS TO STRESS1. The graphical and mathematical definitions of continuity, and the basic principles.2. Examples of discontinuity.3. The Intermediate Value Theorem: mathematical statement, graphical examples, and applied examples.QUIZ QUESTIONS• Text Question: The text says that y = tan x is discontinuous at x = π 2. This would seem to contradictTheorem 7, which states that trigonometric functions are continuous at every number in their domains.Does it? Why or why not?Answer: It does not; tan x is indeed continuous at every point in its domain, but x = π 2is not in itsdomain.• Drill Question: Assume that f (1) =−5, and f (3) = 5. Doestherehavetobeavalueofx, between 1and3,suchthat f (x) = 0?Answer: No, there does not. Only if the function is continuous does the IVT indicate that there must besuch a value.MATERIALS FOR LECTURE• Discuss the idea of continuity at a point, continuity on an interval, and the basic types of discontinuities.• Note that the statement “ f is continuous at x = a” is implicitly saying three things:1. f (a) exists.2. lim f (x) exists.x→a3. The two quantities are equal.To show that all three statements are important to continuity, have the students come up with exampleswhere the first holds and the second does not, the second holds and the first does not, and where the firsttwo hold and the third does not. Examples are sketched below.yyyxxx• Some students tend to believe that all piecewise functions are discontinuous at the border points. Examine⎧2⎪⎨ π x if x < π 2the function f (x) = sin x if π 2 ⎪⎩≤ x ≤ 2π at the points x = π 2and x = 2π. This would be a good2π if x ≥ 2πtime to point out that the function |x| is continuous everywhere, including at x = 0.69