(5) Show that if f is a monotonically increasing function on [4:1, 1;] then f has at most countably many discontinuities. Show the same is true if f : R ) R is an increasing function. (6) Let f" : R. r R be a sequence of functions. Suppose there exists M 2 0 such that |f,,(x)| 5 M for all x E R and n E N. Let A. C R be a countable collection (of points). Show there exists a subsequence (nk such that for all a e A. the sequence fatal) converges. (7) Let . : JR > JR be a sequence of functions such that: o For each n E N, f" is a monotonically increasing function. - There exists M 2 0 such that |f,,(x)l s M for all x 6 IR. and n e N. Show that there exists a subsequence (g); and a function f : R R such that lim six) = f(x) jDOO for all x e R. Hint: First use the previous exercise with A. = Q, dene f(q) = limb\