Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master

SUBJECT

Code
Subject
MMA1016 Topics of Mathematical Analysis I (for teachers education)
Section
Semester
Hours: C+S+L
Category
Type
Didactic Mathematics
1
2+1+0
speciality
compulsory
Teaching Staff in Charge
Assoc.Prof. DIACONU Adrian, Ph.D.,  adiaconumath.ubbcluj.ro
Aims
This course aims to present some special topics of mathematical analysis concerning the sequences and series of real numbers, as well as several important classes of functions.
Content
- Sequences: the limit inferioar and limit superior of a sequence of extended real numbers and their role in the existence of the limit, the set of lmit points of a sequence, Kronecker and Dirichlet Theorems concerning the approximation of real numbers by rational numbers; sequences given by liniar or nonlinear reccurence relations; Toeplitz Theorem and some of its corollaries (Stolz-Cesaro and Cauchy Theorems).
- Series of real numbers: Cauchy and Riemann Theorems concerning the rearrangement of the terms of a series; Abel, Cauchy and Mertens Theorems concerning the Cauchy product of two series.
- Semicontinuous functions: the characterization of semicontinuity by means of sequences, the limit inferioar and limit superioar of a function at a point and their relationship with semicontinuity.
- Uniformly continuous functions between two normed spaces: the characterization of uniform continuity by means of sequences, the relationship between uniformly continuous functions and other important classes of functions (Lipschitz and Hölder continuous functions).
- Functions having the Darboux property and functions admitting primitives.
- Convex functions of one or several real variables: characterizations of convex functions; regularity properties of convex functions; generalized convexity; remarkable inequalities.
References
1. BORWEIN, J.M., LEWIS, A.S.: Convex Analysis and Nonlinear Optimization. Theory and Examples. CMS Books in Mathematics, Springer, 2000.
2. BRECKNER, B.E., POPOVICI, N.: Probleme de analiză convexă în R^n. Casa Cărţii de Ştiinţă, Cluj-Napoca, 2003.
3. COBZAŞ, ŞT.: Analiză matematică (Calcul diferenţial). Presa Universitară Clujeană, Cluj-Napoca, 1997.
4. MEGAN, M.: Bazele analizei matematice. Vol. I + Vol. II, Editura EUROBIT, Timişoara, 1997. Vol. III, Editura EUROBIT, Timişoara, 1998.
5. NICULESCU, C.P., PERSSON L.-E.: Convex Functions and Their Applications. A Contemporary Approach. Springer, 2006.
6. ROBERTS, A.W., VARBERG, D.E.: Convex Functions. Academic Press, 1973.
7. RUDIN, W.: Principles of Mathematical Analysis. 2nd Edition, McGraw-Hill, New York, 1964.
8. SIREŢCHI, GH.: Calcul diferenţial şi integral. Vol. 1: Noţiuni fundamentale. Vol. 2: Exerciţii, Editura Ştiinţifică şi Enciclopedică, Bucureşti, 1985.
9. TRIF, T.: Probleme de calcul diferenţial şi integral în R^n. Casa Cărţii de Ştiinţă, Cluj-Napoca, 2003.
Assessment
Continuous evaluation (contributes 20% to the assesment), written and oral exam (contributes 80% to the assesment).
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject