Math 316 Real Analysis I assignments This page requires Internet Explorer 6+ MathPlayer or Mozilla/Firefox/Netscape 7+.

## Martin Flashman's Courses MATH 316 Real Analysis I Spring, 2013 MTRF 2:00-2:50 NR 201 Assignments (Tentative- WORK IN PROGRESS) Watch for actual DUE Dates before starting work! Assignments are due by 5:00 pm on the due date.

No credit for answers without neatly organized work!
Optional Problems will be graded for bonus assignment points.
Due Date
Problems
* indicates the problem will be collected
Optional
Friday
April 19
SOS Ch 3 pp 49,51, 53
Ch 11 pp279-282

Uniform continuity

Beginning series

Tests for convergence
0. SOS 3.10a, 3.13, 3.21, 3.29, 3.30; 11.2-11.3,
1*. Let f(x) = 5x -2 for all  x in R.
i)Use the epsilon-delta definition of continuity to prove f is continuous on R.
ii)Use the epsilon-delta definition of uniform continuity to prove f is uniformly continuous on R.
2*. Suppose f: R ->R and g:R->R are continuous functions. Let h(x) = f(g(x)) for all x in R.
i) Use the epsilon-delta definition of continuity to prove h is continuous on R.
ii) Use the topological characterization of continuity to prove h is continuous.
3.* Suppose A sub R. Define I(A) = cup{O sub R: O is open and O subA}.
a.* Find (and justify your result) I([4,5]).
b.* Prove: i) I(A) sub A. ii)If O is an open sub set of A then O sub I(A). iii)If A is an open set then I(A)=A.
c.* Prove: Let C denote the Cantor set. Prove I(C)=O/.

Friday
April 12
SOS Ch 12 p324, 326
SOS Chapter 11 p279-283
Infinite series and function.

Connectedness

Continuous functions and connected sets

The Intermediate Value Theorem (revisited)

Compact sets(topological definition)

Uniform continuity

The real line is uncountable

The Cantor Set = C. Review the definition of the Cantor Set .See Integration Notes from Week 10
1*. Suppose a_n is a sequence of numbers in the Cantor Set. Prove there is an element b in the Cantor set where b ne a_n for any n in N. [This the Cantor Set is "uncountably infinite".]

2*.
Suppose f : [0,1] -> R with
f(x) = 1 for  x in C  and f(x) = 0 for x in [0,1]-C.
a) Prove f is continuous for [0,1]-C.
b) Prove f is Darboux integrable and \int_0^1 f = 0.

3*. Prove that [0,1] - C  is an open set.

4*. Suppose a,b in R, a ne b. Prove there exist open sets, U and W with  a in U, b in W and U cap W = O/.

Friday, April 5

### Chapter 4: Continuous Functions on R

Limits of functions

Limits of functions in terms of limits of sequences

Continuous functions
Open and closed sets in metric spaces

Continuity and open sets

Properties of open and closed sets

Constructing and recognizing open and closed sets
0. SOS  5.9, 5.13, 5.28
1*. SOS 5.44
2*. SOS 5.46
3*. SOS 5.80
4. Suppose f is a function on [a,b] where for all x,y in [a,b], if x<y then f(x) \le f(y).
a*. If P = {t_0, t_1, ..t_n} is any partition of [a,b], what is L(f,P)  and U(f,P)?
b*. Suppose t_k-t_{k-1} = delta for each k.
Prove U(f,P)-L(f,P) = delta [f(b)-f(a)].
c*. Prove f in Darboux integrable.

Spivak
Ch 13 Integrals
Ch 14 FTofC
Ch 5 Limits
Thursday
March 28
SOS Ch 5
Interactive Real Analysis 7.1 The Riemann Integral

Analysis Web Note:
Chapter 8: Integration

0. SOS  4.25, 4.26, 5.2, 5.3, 5.5, 5.6
Definition: Suppose f : R -> R. We say lim_{x -> oo}f(x) = L if for any unbounded sequence {a_n} with a_{n+1} > a_n  for all n,  lim_{n -> oo}f(a_n) = L.
1 a.* Prove lim_{x -> oo}1/x = 0.
b.* Prove if lim_{x -> oo}f(x) = L and lim_{x -> oo}g(x) = M then lim_{x -> oo}(alpha f(x) + g(x)) = alpha L + M  where alpha in R.
2*. SOS. 5.34
3*. SOS. 5.42
4*. Suppose f : [0,4] -> R and f(x) = 5 for x in [0,2) and f(x) = 7 for  x in [2,4].
Use the definition of the definite integral [ Riemann or  Darboux- your choice] to show that
\int_0^4 f  = 24.

Friday
March 8
SOS Ch.4

Differentiable Functions 6.5
0.  SOS: 4.1, 4.4, 4.5, 4.12, 4.13, 4.15, 4.22, 4.23, 4.25
1.* 4.68
2.* 4.72 [Use the MVT.]
3.* Suppose f is a continuous function on an interval (a,b) and c in (a,b) with f(c) = L >0.
Prove: There is a natural number M > 0 so that for  all x in (c-1/M, c+1/M), f(x) > 0.
4. a. Prove: If  f(x) = 1/x for x \ne 0 , then f  '(x) = - 1/{x^2} for x \ne\ 0. [Not to be submitted.]
b.* Use the chain rule, the product rule, and part a to prove the quotient rule.
5. Let D =[-2,-1] uuu (2,3] and  f(x) = x^2 for  x in D.
Let E = [1,9] and g(x) = -sqrt(x) for x in [1,4] and g(x) = sqrt(x) for x in (4,9].
a.* Prove: g (f (x)) = x  for all x in D and  f (g (y)) = y for all y in E, so g = f^{-1}.
b.* Explain why f  is continuous on D , but g is not continuous on E. [Give details to show why g is not continuous at 4.]
6. Suppose f  is continuous on an interval [a,b] and for all x, y in [a,b] if x <y then f(x)<f(y).
a.* Prove: f([a,b]) = {y in R: y = f(x) for some x in [a,b]}= [f(a), f(b)]. [This is an equality of sets.]
b.* Prove: There is a unique function g: [f(a),f(b)] -> [a,b] so that g (f (x)) = x  for all x in [a,b] and  f (g (y)) = y  for all y in [f(a),f(b)].
c.* Prove: The function g  described in part b is continuous on [f(a), f(b)].

.
Thursday
Feb 28
SOS:
Ch.3.

Continuity of Real functions

The boundedness theorems
Images of intervals
0. SOS:  2.16, 2.18; 3.3, 3.6, 3.16
1. * Prove: If  {a_n} is a sequence with a_n > 0  for all n and lim_{n -> oo}a_n = L, then L >= 0.
2.* Prove that if a subsequence of a Cauchy sequence converges, then the original Cauchy sequence converges.

3.* Suppose the sequence {a_n}  is bounded and monotonic and {b_n} is a sequence with |a_n - b_n| = (b_0-a_0)/ {2^{n}}. Show that {b_n} converges, and lim_{n -> oo}a_n = lim_{n -> oo}b_n .

4.a*. Prove that   1/{n+1} < ln(n+1)- ln(n) <  1/ n .
b. * Let a_n = 1 + 1/2 + 1/ 3  + ... +1/n - ln(n).
Show  that a_n >= 0  and a_n > a_{n+1} for all n.  [thus there is a limit for the a_n which is called Euler's constant and denoted gamma.]
5.a* Give a definition of piecewise continuous based on the definition of continuity for a set D using sequences.
b*Use your definition to prove that the function h where h(x) = 1  for x >= 0 and h(x) = 0 for x<0  is piecewise continuous.

Thursday
Feb 21
SOS:
Ch.2 p25-27.

Cauchy sequences

Limits of functions
The epsilon-delta definition

Continuity of Real functions

Some horrible functions
0.   2.13, 2.14, 2.16, 2.18, 2.22, 2.24, 2.35
1. Suppose f and g be real valued functions which are continuous on (-oo,oo) and  f (0) = g(0).
a.* Prove that the function defined by h(x) = f(x) for x <= 0 and h(x) = g(x) for x > 0 is continuous everywhere.
b.* Prove that the absolute value function | x | is continuous everywhere.
c.* If f is a continuous function on R, prove that the function | f(x) | is also continuous on R.
d.* If f and g are continuous functions prove that the function M(x) = max{f (x), g (x)} is also continuous.
[Hint: Prove that max(a, b) =((a + b) + |a - b|)/2 for any real numbers a and b.]
e. Prove that the function m(x) = min{f (x), g (x)} is continuous if f and g are.
2.a.* Use the geometry of the sine function on the unit circle to prove that | sin x - sin y | <= | x - y |
b. * Prove that the function f(x) = sin x is continuous everywhere.
c. Prove that the function cos x is continuous everywhere.
3.* Suppose f :[0,1] -> [0,1] is a continuous function. Prove there is some a in [0,1] where  f(a) = a.

Thursday
Feb 14

SOS: p6 (Bounds)
Ch.2 p25-26.

Axioms for the Real numbers

Sequences of Numbers

The intermediate value theorem
0.  SOS: 1.21(a,b); 1.64(a,b); 1.65(a,b);  2.20(a,b) ; 2.21 (a,b); 2.22
*1. Suppose A and B are nonempty sets of numbers and that for any x in A and any y in B,
x <=y.
a) Prove that the LUB (A) <= y  for all y in B
b) Prove that LUB(A) <= GLB( B).
*2. Suppose A and B are nonempty sets of numbers that are bounded above. Let A+B denote the set of numbers that can be expressed as a+b where a is in A and b is in B.
Prove: LUB (A + B) = LUB(A) + LUB(B)
*3 .a) Suppose I_1 = [a_1,b_1], I_2 = [a_2,b_2], ... is a sequence of close intervals with a_n<b_n,  a_n<=a_{n+1) and  b_n >= b_{n+1} for all n.
Prove: There is a number c that is in all the intervals.
b) Give a counterexample to show that the statement in part a is not true if the intervals are open instead of closed.
Spivak Ch 8
especially Proof of Theorem 7.1

SOS: 1.34 on Dedekind Cuts
Thursday
Feb. 7
SOS: Chapter 3 THEOREMS ON CONTINUITY p52
*1.Give a statement the Extreme Value Theorem with examples illustrating how it can be false if one of the hypotheses is not satisfied.
*2. Suppose f is a continuous function on a domain D that is the union of a finite number of intervals of the form [a_j, b_j] with a_j < b_j for j = 1 to n and a set of real numbers { c_k: k = 1 to m}.
Prove that there are numbers p and q  in D, where f(p) <= f(x) <= f(q) for all x in D.
*3. Suppose f is a continuous function on a domain D that is the union of a finite number of intervals of the form [a_j, b_j] with a_j < b_j for j = 1 to n and set of real numbers { c_k: k = 1 to m}. Using the Intermediate Value Theorem and the Extreme Value Theorem, prove that f (D) = {y : y = f (x) for some x in D}  is also the union of a finite number of intervals of the form [r_i, s_i] with r_i < s_i for i = 1 to  n' and set of real numbers (possibly empty){ t_l: l = 1 to  m'}.
Spivak: Ch 7 pp 120-122
Thursday
Jan 31
SOS:
Chapter 1
Up through POINT SETS, INTERVALS (p5)

Chapter 4  MEAN VALUE THEOREMS.(p77-78)
[Proofs can be found in the solved problems 4.19, 4.20(Note error in F.)]

*1.Give a statement and proof of the Mean Value Theorem and of Rolle's Theorem. [You may use other results for your proof. If you use Rolle's theorem to prove the MVT- prove Rolle's Theorem without the MVT!]
*2. Prove: Rolle's Theorem is equivalent to the Mean Value Theorem.
*3. For any real number x let |x| be the absolute value of x, i.e., |x| = x when x>0,|x| =-x when x< 0, and |x| = 0 when x =0.
Prove: For any real numbers a and b,
|a+b| <= |a| + |b|.
*4. Prove the following:
(i) |a-b| <= |a| + |b|.
(ii) |a| - |b| < = |a-b|.
(iii) |( |a| - |b| )| <= |a-b|.
*5. Suppose m(x,y) = |y-x|  for and  y real numbers.Prove that for any real numbers, a,b, and c:
(i)  m(a,b) =  0 if and only if  a=b.
(ii)  m(a,b) = m (b,a).
(iii) m(a,b) < = m(a,c) + m(b,c)
(iv) If  m(a,b) = m(a,c) + m(b,c)  then  either  a= c,
b=c,  a<c< b, or  b<c<a.
Spivak: Chapter 1; pp190-192
Tuesday
Jan. 29

### The About Materials  from  Interactive Real Analysis  by Bert G. Wachsmuth

Introduction from A first Analysis course by John O'Connor

Resource Notes. Chapter 0. Introduction  by M. Flashman Available on Moodle.

SC 0.B1  Numbers [on-line]
SC 0.B2 Functions [on-line]
1. Make a list of 5 theorems that you feel are critical to understanding and using the material you learned in the first year of calculus.
Be sure to state clearly the theorem with all its hypotheses and conclusions.
2. For each of the 5 theorems you gave, provide at least one example of how the theorem can fail if only one of the hypotheses is not satisfied.

Future possible problem(s) Definition: Suppose f:D->R . We say f is an open map  if  for any open set O,  there is a set hatO where f(O cap D)= hatO cap f(D).
1. For each of the following functions  f determine if f is an open map. If it is - prove it, if not give an example that shows why f is not an open map.
a. f:R->R, f(x) = x^2
b. f: R->R, f(x) = sqrt(|x|)
c. f: R->R, f(x) = 0
d.f: R->R, f(x) = 5x + 2
e. f: [1,2]->R, f(x) = x^2.
2. Suppose f : R ->R is a 1:1 function: Prove: f is an open map if and only if f^{-1} is a continuous function.