3 Advanced Calc. Questions?

Question

1. Give an example of an open cover of (-1, 1] which does not have a finite subcover.

2. Give an example of a function f: (A U B)-->R such that f is uniformly continuous on A and B but not on (A U B). U denotes union and R denotes the set of real numbers.

3. Give an example of two uniformly continuous functions f,g : D -- > R such that f*g is not uniformly continuous. R denotes the set of real numbers.

Answer 1

1: {(x, ∞): x∈(-1, 1)}

2: Let A=Q and B = R\Q (Q is the set of rational numbers). Then let f:R → R = {1 if x∈Q, 0 if x∉Q}. Then f is uniformly continuous (indeed, it is constant) on both A and B, but on A∪B it is nowhere continuous, let alone uniformly continuous.

3: Let D = R, and let f(x) = g(x) = x. Then f and g are both uniformly continuous, but (fg)(x) = x² is not uniformly continuous on R.