19) Prove or find a counter example to the following statements: a) f is continuous on (a,b) implies that for any (XR)->C, (f(x)) converges b) f is uniformly on [a, b] & [b, c] implies that I is continuous on [a, c] f

1. Suppose f is continuous on R and f satisfies f(x) + f(2x) = 0 for all x E R. Prove that f = 0 on R.

3. Assume that f(x) is continuous on [a, b] and x₁,x2,...,n are arbitrary numbers taken from [a, b]. Show that there exists c E (a, b) so that f(c) 1 · [ƒ(x1) + ƒ(x2) + ... + f(xn)]. n == Hint. Use Intermediate and Extreme Value Theorems.

Use definition of uniform continuity to prove that f(x) is uniformly continuous on [0,infinity)

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].)