TRINITY COLLEGE DUBLIN

Analysis 1 for JF Mathematics, JF Theoretical Physics and JF & SF Two-subject moderatorship

This syllabus may be updated slightly for this year. R. Timoney will lecture the first half of the course and Dr. J. Stalker the remainder.

Notes for Chapter 0 (introduction to the course, motivation for taking care over the real number system) can be found here as a pdf file.
Notes for Chapter 1 (the axioms for the real number system) can be found here as a pdf file. Contents include field axioms, order axioms, triangle inequality, least upper bound principle, application to show 2 has a square root in the real numbers.
Notes for Chapter 2 (basic theory of limits of sequences) can be found here as a pdf file. Includes monotone sequences, Bolzano-Weierstrass theorem, notion of a subsequence. Application that every decimal represents a real number.
Notes for Chapter 3 (limits of function) can be found here as a pdf file. Definition using sequences, basic properties, equivalent epsilon-delta, definition of continuity using epsilon-delta, equivalent fomulations using limits or sequences, basic properties of continuity, one-sided limits.
Notes for Chapter 4 (theorems about continuous functions) can be found here as a pdf file. Compactness, Heine-Borel, continuous functions on finite closed intervals have max values, Intermediate Value Theorem, continuous images of intervals, applications to polynomials. Monotone functions, injective and continuous on an interval implies continuous inverse. Derivatives of inverse functions. Rational powers.

See Dr. Stalkers web site for additional notes.

Exercises 1 [Due Friday October 29th, 2004.]: Practice using the axioms for the real numbers (ordered field axioms only).
Exercises 2 [Due Friday November 26th, 2004.]: Inequalities. Upper and lower bounds. Every real number is a decimal.
Exercises 3 [Due Wednesday December 8th, 2004.]: Limits of sequence and functions.
Exercises 4 [Due Wednesday January 26th, 2005.]: One and 2-sided limits of functions. Compact.

The christmas test with solutions is here.

Exam papers set in previous years are available in PDF format on the TCD local web site.

Updated 5th May, 2005
Richard M. Timoney