Math 104: Homework 7-
1. (a) Let S be a subset of R, and let f : S → R and g : R → R be uniformly continuous functions. Prove that the composition g ? f : S → R is uniformly continuous.
(b) Let f and g be two uniformly continuous functions from S to R. Prove that f + g is uniformly continuous.
(c) Show that there exist uniformly continuous functions f and g from S to R such that the multiplication f · g is not uniformly continuous.
2. Let f be a real-valued continuous function. Recall that for any subset S ⊆ R, then f(S) is defined as {f(x) : x ∈ S}. Suppose that f(I) is open for any open interval I. Prove that f is monotonic.