Math 104, Section 6 Fall 2003SarasonSUGGESTED EXERCISESPages 260–261, Exercises 33.3(a), 33.4, 33.7, 33.8, 33.9, 33.12, 33.14, plus Exercises 1–12 below.1. Let C1and C2be connected subsets of a metric space such that C1∩C2= ∅ and¯C1∩C2 = ∅.Prove C1∪ C2is connected.2. Let A and B be connected subsets of Rk. Prove the setA + B = {a + b : a ∈ A, b ∈ B}is connected.3. Prove the setC = {(x, y, z) ∈ R3: z2= x2+ y2}is connected.4. Prove that, for k ≥ 2, the setSk−1= {x ∈ Rk: x =1},the unit sphere in Rk,isconnected. (Suggestion: Use induction on k.)5. Prove there is no one-to-one continuous function of R2onto R.6. Let I ⊂ R be an open interval, and let the function f : I → R be differentiable to secondorder, with fcontinuous. Prove that, for x in I,f(x)=limδ→0f(x + δ)+f(x − δ) − 2f(x)δ2.7. Let I ⊂ R be an open interval. Let f : I → R satisfylim infx→af(x) − f(a)x − a> 0for all x in I. Prove f is increasing.8. Let f :[a, b] → R be Riemann integrable and let g :[a, b] → R be bounded. Prove U(f + g)=U(f)+U(g) and L(f + g)=L(f)+L(g).9. Let f :[a, b] → R be bounded. Assume f is Riemann integrable on [c, b] for every c in (a, b).Prove f is Riemann integrable on [a, b].10. Prove that the characteristic function of the Cantor set is Riemann integrable and that itsintegral is 0.11. Let f :[a, b] → R be bounded. Prove that for f to be Riemann integrable it is necessary andsufficient that, for every ε>0, there are continuous functions g :[a, b] → R and h :[a, b] → Rsuch that g ≤ f ≤ h andbah −bag<ε.12. Let f :[a, b] → R be differentiable, with |f|≤M. Fix a positive integer n, and let tk=a +kn(b − a) for k =0, 1,...,n,sothat (t0,t1,...,tn)isthe partition of [a, b] for which eachsubinterval has lengthb − an.Foreach k let xk=tk−1+ tk2be the midpoint of [tk−1,tk], andlet Rn=nk=1f(xk)(tk− tk−1). Prove that the absolute value of the difference between theRiemann sum Rnandbaf is bounded byM(b −