The analysis preliminary exam is designed to test students' knowledge of basic concepts associated with the study of functions of real variables and their ability to formulate proofs of assertions about these basic concepts:
Further required material is in the following recommended exercises. You may need to consult other books to solve some of the exercises.
The primary references are:
We also recommend the following supplemental texts, which contain the same topics covered from different vantages and some at a greater depth.
Rudin's book is the best source, but many problems from it are too long to be useful practice problems for the prelim exam. Shorter problems from other references are more appropriate. The student exam paper must be written in a way that the evaluators can understand clearly the student's line of argument so that the correctness of the argument can be decided. Please do not write incorrect statements, hoping for partial credit. The evaluators will not fill in missing arguments, make educated guesses about what the student had in mind, or patch a solution from correct statements mixed with incorrect ones. The expected style of the solutions is essentially the same as as seen in graduate texbooks, in monographs, or in papers. In particular:
To prepare for the exam, students are strongly advised to solve as many problems from the recommended texts as possible, in particular the recommended exercises above, try to find counterexamples to theorems when some assumptions are dropped, and practice writing their solutions in the required style. The classes MATH 5070, Applied Analysis; MATH 4310, Introduction to Real Analysis I; and MATH 4320, Introduction to Real Analysis II, can help with preparation for this exam. However, students should note that these classes are not for PhD students only, so the grading of students' work in these classes cannot reflect the standards of mathematical accuracy and expression expected on this exam. Further, these courses do not cover all of the required material.