This is a multiple choice question.

Let I be an interval and let fn : I -> R for each n E N. We say that the sequence { fn} of functions converges pointwise to f : I - R on I if lim fn(x) = f(x) n -+ 00 for every x E I. Which of the following statements are true? a) If fn(a) = sin(" ) for each n E N, then {fn} converges pointwise to the function f(x) = 0 on [0, 1]. b) If fn(x) = x for each n E N, then { fn} converges pointwise to a function f on [0, 1]. c) If fn is a continuous function and {f} converges to f on [0, 1], then f is also continuous on [0, 1]. d) Assume that f is differentiable on R. Let In(x) = (at 2. Then {on} converges pointwise on R to f'. n e) If fn is a sequence of continuous functions that converges pointwise to a continuous function f on [0, 1], then lim n -too Jo fn(x)dx = / f(x)dx