7. Let E <:;;; R. A function j : E ~ R is said to be increasing on E if and only if Xl, x2 E E and Xl < X2 imply j(XI) ::::; j(X2). Suppose that j is increasing and bounded on an open, bounded, nonempty interval

(a, b).

(a) Prove that j (a+) and j (b-) both exist and are finite.

(b) Prove that j is continuous on

(a,

b) if and only if j is uniformly continuous on

(a, b).

(c) Show that

(b) is faise if j is unbounded. Indeed, find an increasing function g : (0, 1) ~ R that is continuous on (0,1) but not uniformly continuous on

(0,1).