We expect that the student has approved an undergraduate sequence in Analysis, this is usually achieved by taking the undergraduate courses MATE 4051, Advanced Calculus I and MATE 4052, Advanced Calculus II or by taking an upper level undergraduate course in Analysis.

  • Real numbers. Order axioms, axiom of completness.
  • Basic topology in R: Compact sets, Bolzano-Weierstrass and Heine-Borel Theorems, connected sets.
  • Sequences and series: monotone sequences, convergence tests, limit sup and limit inf, Cauchy sequences.
  • Continuity: Basic definition, uniform continuity, types of discontinuities, limit of sequence of functions.
  • Differentiation: Mean Value Theorem, Taylor Theorem, L'Hôpital.
  • Riemann-Stieltjes integral: definition, basic properties, Fundamental Theorem of Calculus, Mean Value Theorem integral form. Functions of bounded variation.
  • Sequences and series of functions. Absolute and uniform convergence, integration and differentiation, power series, Taylor series, radius of convergence.
  • Functions of several variables. Differentiation, Inverse Function Theorem, Implicit function Theorem, Rank theorem.
  • Integration in the Lebesgue sense. Measurable sets and measurable functions, Fatou’s Lemma, Monotone convergence Theorem and Dominated convergence Theorem.


  • Mathematical Analysis, T. M. Apostol, Second Ed., Addison-Wesley, 1974.
  • Measure Theory, D.L. Cohn, Birkhäuser, 1994.
  • The Elements of Real Analysis, R. G. Bartle, Second Ed., Wiley & Sons, 1976.
  • Real Analysis: Modern Techniques and Their Applications, G.B. Folland, Wiley & Sons, 1999.
  • Advanced Calculus, W. Fulks, Third Ed., Wiley & Sons, 1978.
  • Elementary Analysis: The Theory of Calculus, Ross, A. Kenneth, Springer-Verlag, 1980.
  • Principles of Mathematical Analysis, W. Rudin, McGraw-Hill, Third Ed., 1976.
  • Introduction to Real Analysis, R. G. Bartle and D. R. Sherbert, Wiley & Sons, Third Ed., 2000.
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