Suppose that α ∈ R, that E is a nonempty subset of R, and that f, g : E → R are uniformly continuous on E.
a) Prove that f + g and αf are uniformly continuous on E.
b) Suppose that f,g are bounded on E. Prove that fg is uniformly continuous on E.
c) Show that there exist functions f,g uniformly continuous on R such that fg is not uniformly continuous on R.
d) Suppose that f is bounded on E and that there is a positive constant ε0 such that g(x) ≥ ε0 for all x ∈ E. Prove that f/g is uniformly continuous on E.
e) Show that there exist functions f,g, uniformly continuous on the interval (0,1), with g(x) > 0 for all x ∈ (0, 1), such that f/g is not uniformly continuous on (0,1).
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