Let ⊂ 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 / is continuous on (a, b) if and only if / is uniformly continuous on (a,b).
c) Show that b) is false if / is unbounded. Indeed, find an increasing function g : (0, 1) → R which is continuous on (0,1) but not uniformly continuous on (0,1).