Chapter 5.3, Problem 116E

Continuity at the endpoints Assume that f is continuous on [a, b] and let A be the area function for f with left endpoint a. Let m* and M* be the absolute minimum and maximum values of f on [a, b], respectively.

1. a. Prove that m*(x − a) ≤ A(x) ≤ M*(x − a) for all x in [a, b]. Use this result and the Squeeze Theorem to show that A is continuous from the right at x = a.
2. b. Prove that m*(b − x) ≤ A(b) − A(x) ≤ M*(b − x) for all x in [a, b]. Use this result to show that A is continuous from the left at x = b.