1) Let f, g be defined on R and let c in R. Suppose that lim f = b and that g is continuous at b. Show that lim g 0 f = g(b)
R: real numbers
g 0 f means composition of f and g
2) Let A = [0, 1) U (1,2]. Let B = [0, 1] U [2, 3]. Does the conclusion of the maximum-minimum theorem always hold for a function f: A -> R, g: B ->R that is continuous on A, on B respectively? Prove or give a counterexample.
U: means union
Max-min theorem: f: [a, b] ->R continuous on [a, b]. Then f has an absolute max and an absolute min on [a, b]
Maximum-Minimum Theorem, Limits, Continuity and Function Composition are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.