Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

Let f(x) be differentiable for all x on R. Suppose that f(0) = -2  and that f'(x) ≤ 1 forall x on R. Use the Mean Value Theorem to show that f(4) ≤ 2.

Suppose f is continuously differentiable on an interval (a, 6). Prove that on any closed subinterval [c, d] the function is uniformly dif- ferentiable in the sense that given any 1/n there exists 1/m (inde- pendent of ro) such that |f(x)- f(xo)-f"(x0)(x-x0)| < |x- xo|/n whenever r – xol < 1/m. (Hint: use the mean value theorem and the uniform continuity of f' on [c, d].)