REAL ANALYSIS 1-MA 206-(4 credits)-Fall 2012

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The Cantor Function

INSTRUCTOR: Dr. Botsko

TEXT: An Invitation to Real Analysis (3rd edition) by Michael W. Botsko

PREREQUISITE: MA 114

COURSE CONTENT:

We will cover the concepts found in chapters 1, 2, 3, 4, and 5 of the above text. A list of topics that we will study includes:

  1. The real number system
  2. Limits of sequences and functions
  3. Monotone sequences and functions
  4. Cauchy sequences
  5. Limit superior and limit inferior
  6. Continuity and uniform continuity
  7. The Full Cover Theorem and continuity
  8. The derivative
  9. Sets of measure zero
  10. Absolute continuity and derivatives
  11. The Mean Value Theorem and L'Hospital's Rule
  12. Taylor's Theorem and optimization
  13. The Riemann Integral
  14. Lebesgue's Theorem
  15. A generalized fundamental theorem of calculus
  16. Mean value theorems for integrals
  17. The Riemann-Stieltjes Integral
  18. Products of derivatives
  19. The Lebesgue Integral
  20. The Riemann-complete Integral

Real Analysis I emphasizes the study of real functions of a single variable. In Real Analysis II, offered next semester, we will consider real and vector valued functions of several variables.

OBJECTIVES:

To know the basic definitions and fundamental theorems of Real Analysis, to be able to prove simple propositions and to develop proficiency in applying the problem-solving techniques treated in the course, to be able to connect Real Analysis with other branches of mathematics. Class discussions and exams will be used to assess the level to which these objectives are being attained.

EXAMS:

There will be three exams and a final and you will be given a week's notice for each exam.

EARLY PERFORMANCE GRADES:

You will be assigned an early performance (near mid-term) grade around the middle of October which will be based on your performance on the first exam.

GRADING POLICY:

The final will count as 1/4 of your grade as will each of the three in class exams. The grading system will be according to the current SVC bulletin.

CLASS ATTENDANCE:

Class attendance is very important in an upper division mathematics class. Please make a sincere effort to be present at every class. If for some reason class is cancelled, an announcement will be posted on the Blackboard site for this course.
ACADEMIC HONESTY:

"Saint Vincent College assumes that all students come for a serious purpose and expects them to be responsible individuals who demand of themselves high standards of honesty and personal conduct. Therefore, it is college policy to have as few rules and regulations as are consistent with efficient administration and general welfare.
Fundamental to the principle of independent learning and professional growth is the requirement of honesty and integrity in the performance of academic assignments, both in the classroom and outside, and in the conduct of personal life. Accordingly, Saint Vincent College holds its students to the highest standards of intellectual integrity and thus the attempt of any student to present as his or her own any work which he or she has not performed or to pass any examinations by improper means is regarded by the faculty as a most serious offense."
Saint Vincent College 2009-2011 Bulletin, page 20..
CLASSROOM ETIQUETTE:

An essential characteristic of Saint Vincent College is the dignity and civility with which students and instructors conduct themselves both inside and outside the classroom.  All students share in the responsibility of making the classroom a positive place to learn.  Attendance is more than just being in the classroom; students are expected to be prepared and attentive.  Cell phones, pagers, and other electronic devices should be turned off when students are in the classroom.

DISABILITY STATEMENT: 

Students with disabilities who may be eligible for academic accommodations and support services should please contact Mrs. Sandy Quinlivan by phone (724-805-2371), email (sandy.quinlivan@email.stvincent.edu) or by appointment (Academic Affairs-Headmaster Hall).  Reasonable accommodations do not alter the essential elements of any course, program or activity.

OFFICE HOURS:

I will be in my office (4th floor of Alfred Hall) at the following times during the week. When coming for help be sure to bring along your textbook, notes, and your efforts at solving the problems.

Monday: 9:30 to 10:30 and 3 to 4

Tuesday: 10:30 to 11:30

Wednesday: 9:30 to 10:30

Friday: 9:30 to 10:30

Note: The function at the top of this page, The Cantor Function, is an example of a continuous increasing function whose derivative is 0 almost everywhere. It is a function of bounded variation but it is not absolutely continuous. To see the animation again, simply click the "refresh" or "reload" icon at the top of the screen. If you would like to see the Cantor Function constructed frame by frame, click here .

Click here for a nice interactive page in Real Analysis.

Blackboard (If this link doesn't work, try here)