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

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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/`.

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


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".]

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.
Ch 13 Integrals
Ch 14 FTofC
Ch 5 Limits
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.`

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)].

Feb 28

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.

Feb 21
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`.

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
Feb. 7
*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
Jan 31
Chapter 1

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
Jan. 29

Chapter 1:Analysis WebNotes by John Lindsay Orr  Introduction

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.