Prove that if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is Lipschitz continuous on (−∞, ∞).

Let f : [0, 1] → R be a continuous function on the closed interval [0, 1] and differentiable on the open interval (0, 1). Assume that f(0) = 0 and f’ is an increasing function on (0, 1). Show that g(x) =f(x)/x is an increasing function on (0, 1)

Let f : I → J be a bijective differentiable function froman interval I to an interval J, and let F : I → R be an anti-derivative of f. Find an explicit expression, in terms of f, f-1 and F, for an anti-derivative of f-1: J → I. [Both substitution and integration by parts may come in handy.]

g(x, y) = sin(xy) xy 1, Prove that g is continuous on R². 9 if (x, y) = (0,0) if x = 0 or y = 0.