Project Descriptions
Real Analysis 302
The following descriptions are meant as some suggestions to get you started on your
projects. You are free to modify them as you wish. Remember the goal is to have you
independently learn and present some analysis! So find something that intrigues you and
become an expert on the topic! Once you have chosen a project and started working, you
will meet with Prof. Donnay to discuss your project in more detail.
#1 : Continuous but Nowhere Differentiable Function
Present to the class an example of a continuous function that is nowhere differentiable!
It involves taking a sequence of continuous functions fn(x) and showing they converge
uniformly to a limit function f(x).
See Appendix E in Bartle; Example 2 of Section 6.3 and exercise 13 in Section 4.1 in
Reed; and Sect. 9.7 in Goldberg, This function is an example of a fractal .
Try applying these ideas to the Koch Snowflake and other interesting fractal shapes.
#2 Space Filling Curves
Demonstrate how the idea of uniform convergence of functions can be applied to show
the existence of a continuous function f:[0,1] -> R2 whose image (also known as range) is
a 2-dimensional region of the plane! So for example, it is possible to construct such a
function that passes through every single point of a (solid) triangle in the plane.
See Appendix E of Bartle. There are many other sources where you can find this
construction. A nice source is Basic Topology by M.A. Armstrong, pp 36-37. I’ll put the
entire Armstrong book on reserve. If you choose to use Armstrong, I suggest:
Describe the sequence of continuous functions fn; (Note that E2 is just R2 equipped with
the standard metric.)
Prove that {fn} form a Cauchy sequence in the sup norm and thus converge uniformly;
Show that the limiting function f is continuous;
Show that every point in the triangle is in the image of f:
Show that every point of the triangle is a limit point of f; (Here, explain that p being a
limit point means that every open ball around p must contain at least one point of the set
f[0,1] – {p}. More details about this generalized version of limit points can be found on
p. 29 of Armstrong.)
Cite the theorem that says that the continuous image of [0,1] must be a closed subset of
R2 ;
Cite the theorem that says that a set is closed if and only if it contains all its limit points;
(See p. 29 of Armstrong for the statement of this theorem.)
You may also be able to find some nice references from the Web for this project! There is
a video on the topic of space filling curves available through the library.
#3 Newton’s Method
In this project, you will explore Newton’s method which is a technique to find zeros of
functions. Reed Section 4.4. (see also Bartle, Section 6.4 p. 189-191). For this project,
you should carefully read 4.4.
Explain and illustrate Newton’s method. Notice that Newton’s method produces a
sequence of numbers. What issues arise with a sequence of numbers?
Apply Newton’s method to examples. For instance, find the roots of
f(x) = x4 – 6x2 + 2x + 1. (This is Problem #1 from 4.4.). See also Problem #3 from 4.4.
Explain the statement and poof of Theorem 4.4.1. The proof of this will involve Taylor’s
Remainder 
You could compare Newton’s Method of finding roots with other methods (such as
bisection, or other iteration methods).
Sometimes Newton’s method does not converge; other times you might approach a
periodic orbit rather than a single solution.
#4: Banach Fixed Point Theorem.
Understand and prove the Banach Fixed Point Theorem (sometimes called the
Contraction Mapping Theorem) in the setting of a metric space.
Give examples from Dynamical Systems in which this Theorem applies to show the
existence of a fixed point.
Apply this Theorem, in the context of a metric on the space of functions, to prove the
Existence and Uniqueness Theorem for Differential Equations.
One of the key theorems in the field of differential equations is the Existence and
Uniqueness Theorem: a differential equation of the form y'(t) = f(t, y), where f satisfies
certain conditions, will have a unique solution satisfying the initial condition y(t=t0) = y0.
The proof of this theorem is an elegant application of the Contraction Mapping Theorem.
See Introductory Functional Analysis by Kreyszig Ch 5; specifically section 5.3. Also
Reed, Section 7.1. There is an easy to use computer disk from the Differential Equations
textbook (Differential Equations by Blanchard, Devaney and Hall) that will help you
understand about solutions of differential equations – ask Prof. Donnay about it.
In Math 210, this theorem is often quoted but not usually proved. Now with the
techniques of Real Analysis, we can prove. It.
One considers the complete metric space C(I) of continuous functions defined on an
interval I with the sup metric. One defines a mapping T: C -> C by equation (7.1.2) in
Reed and shows that this map is a contraction mapping so it has a unique fixed point.
This fixed point in the function space is the solution of the differential equation.
Components of this project could include:
for a specific differential equation and initial condition, start with an initial function y0
and then calculate the first few functions in the sequence
yk = T yk-1. You might use Mathematica to help with the calculation and with graphing
the functions. Take the equation y' = 1 + y2, with initial condition y(0) = 0 and the initial
function y0(t) =0. For this equation, you can find the exact solution. Compare the iterates
yn(t) with the exact solution.
Prove Theorem 7. 1 using the language of the Contraction Mapping Theorem (see
problem #5, Sect 7.1 Reed). You could prove the theorem in the simpler case that the
function f only depends on y: for example f(y) = sin(y) (#2, Sect 7.1, Reed). For that
specific example, work through the proof and answer question #2.
Give an example of a differential equation for which the conditions of theorem fail and
show that in such a case one can have a more than one solution to the problem. (Nonuniqueness). Such examples can be found in Differential Equations books such as
Differential Equations by Blanchard, Devaney and Hall that is on reserve for the Math
210 course.
#5 Dynamical Systems and the Quadratic Map.
Study iteration xn+1 = f(xn) where f is the quadratic map: f(x) = ax(1-x ) and examine
how the behavior of the iteration varies as the constant "a" in the formula changes. See
the book A First Course in Chaotic Dynamical Systems by Robert L. Devaney.
Explain how and why the Staircase diagram works. See Figure 2.7.1 in Reed.
Prove the Attracting Fixed Point Theorem: (Roughly stated). A fixed point x0 for a
dynamical system given by xn+1 = f(xn) is attracting if |f’(x0)| < 1 (repelling if |f’(x0)| > 1).
Determine the fixed points of the quadratic map for various values of a. Determine when
the fixed points are attracting or repelling. Determine the period 2 orbits of the map and
when these orbits are stable or unstable.
Learn about the ‘bifurcation diagram’.
Possible directions: Reed Theorem 2.7.1 and 2.7.2. Do numerical investigations of what
happens for values of "a" in the interval [3, 4]. Look for periodic orbits of various
periods.
#6 Numerical Integration and Error Estimates
The Riemann integral is defined as the limit of the Riemann sums. We saw that these
sums can converge very slowly to their limiting value. It can take a very large n value
(small x value) before the Riemann sum agrees with the limiting value (i.e. with the
exact value of the integral) to 5 decimal places. There are a variety of numerical methods
that produce faster convergence to the limiting value: the Trapezoid Rule, the Midpoint
Rule, and Simpson’s Rule.

In this project, you will experiment with the rates of convergence of these different
methods using Mathematica.
There are theorems that give rigorous bounds for how large the error between the
approximation and the exact value can be in terms of the number of intervals N used in
the sum. Your analysis skills will allow you to understand the proofs of these results.
Reed, Section 3.4 and problems from that section. Bartle Section 7. 4, particularly p. 221225, and then the proofs are given in Appendix D. See the problems on p. 225-226. Any
calculus book will give the basics about these methods. For example, Stewart Edition
6E, Section 8.7 – Approximate Integration.
This area of mathematics, using analysis to examine the efficiency of various numerical
methods is called Numerical Analysis.
Project Planning.
Name:
Your top three choices for project in order of preference:
1.
2.
3.
People you would particularly like to work with:
People (if any) you would rather not work with: