Let E ⊆ R. A function f : E → R is said to be increasing on E if and only if x1, x2 ∈ E and x1 < x2 imply f (x1) ≤ f (x2). Suppose that f is increasing and bounded on an open, bounded, nonempty interval (a, b).
a) Prove that f (a+) and f (b−) both exist and are finite.
b) Prove that f is continuous on (a, b) if and only if f is uniformly continuous on (a, b).
c) Show that b) is false if f is unbounded. Indeed, find an increasing function g : (0, 1) → R which is continuous on (0,1) but not uniformly continuous on (0,1).
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