1. Suppose that f is uniformly continuous on [a, b] and uniformly continuous on [b, c]. Prove that f is uniformly continuous on [a, c]. 2. A function f : R → R is said to be periodic if there exists a...

1. Suppose that f is uniformly continuous on [a, b] and uniformly continuous on [b, c]. Prove that f is uniformly continuous on [a, c].

2. A function f : R → R is said to be periodic if there exists a number k > 0

such that f (x + k) = f (x) for all x ∈
. Suppose that f :
 →
 is continuous and periodic. Prove that f is bounded and uniformly continuous on
.