continuous

Can you help the following problems? 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) Note: 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. Note: 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]

Max-Min theorem

Recall: Max-Min theorem: Let f: [a, b] -> R (real numbers) be continuous on [a, b]. Then f has an absolute maximum and an absolute minimum on [a, b] Please see attachment for problem. Thanks in advance for your help

Mean, Median and Mode

2. Find the mean, median, and mode of the following data set: 5 15 9 22 67 42 2 72 81 53 6 70 41 9 42 23

finite or countable collection of disjoint open intervals

(Complete problem found in attachment)

Sequences and Subsets

(See attached file for full problem description)