Let f:[0,1]-> R be bounded on [0,1] and continuous on (0,1). Prove that f is Riemann integrable on [0,1]. Hint: Show that for any epsilon > 0 there exists a partition P of [0,1] such that U(f,P) - L(f,P) < epsilon.
So f is bounded |f(x)|<=M for some M in R. so the sum(k=0,n)[f(x_k)*(t_k - t_(k-1))] <= sum(k=0,n)[M*(t_k - t_(k-1))
I think it is safe to say that f is uniformly continuous on this interval but not sure I can assume that since its not continuous on the closed interval [0,1] and only continuous on open (0,1). Not sure how to use that it is uniformly continuous (if it even is) to prove that it is Riemann integrable.
I'm at a loss on this. Any help is appreciated. Thanks in advance!