AMATH/PMATH 331 Real Analysis, Assignment 2 Due Mon Oct 26
Read Chapters 2 and 3 in the Lecture Notes, work on Problem Set 2 and read the solutions, then solve each of the following problems.
1,ifx= 1 forsomen∈Z+ 1: (a) Define f : [0,1] → R by f(x) = n
Prove that f is integrable.
0 , otherwise. x2 , if x ∈/ Q
(b)Definef:[1,2]→Rbyf(x)= 2x,ifx∈Q.
Prove, from the definition of the upper integral U(f), that U(f) = 3.
2: (a) Define fn : [0,∞) → R by fn(x) = nxe−nx. Find the pointwise limit f(x) = lim fn(x)
n→∞
and determine whether fn → f uniformly on [0, ∞).
(b)Definef :[0,∞)→Rbyf (x)= x . Findthepointwiselimitf(x)= lim f (x)
n n 1+nx2
and determine whether fn → f uniformly on [0, ∞).
n→∞n (c) Define fn : [0, ∞) → R by fn(x) = x+n . Show that (fn) converges uniformly on [0, r]
x+4n
for every r > 0 but that (fn) does not converge uniformly on [0, ∞).
3:(a)Supposethatfn →funiformlyonA⊆Randletg:R→R. Supposethatfis bounded and g is continuous. Prove that g ◦ fn → g ◦ f uniformly on A.
∞
(b) (The Riemann Zeta Function) Define ζ : (1,∞) → R by ζ(x) = 1 . Prove that ζ
is differentiable on (1, ∞). Hint: use the Weierstrass M-Test, together with convergence tests from first year calculus, to show that for all r > 1 the series 1 and −lnn
nx n=1
nx nx both converge uniformly on [r,∞), then apply Theorem 3.16 (Uniform Convergence and
Differentiation).