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

Code	Subject	
MMA0005	Real Functions	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	4	2+2+0	speciality	compulsory	
Mathematics and Computer Science	4	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Assoc.Prof. ANISIU Valeriu, Ph.D., anisiu@math.ubbcluj.ro
Prof. BULBOACA Teodor, Ph.D., bulboaca@math.ubbcluj.ro	
Aims	Learning the fundamental facts in general topology, measure and integartion

Content	Topological spaces, neighborhoods, closure, interior, boundary
Bases and subbases, generated topology, subspace, product spaces, convergence, continuity.
Separation exioms (T1,T2), metric spaces, completeness.
Compactness.
Etension of continuous functions. Baire functions.
Connectedness.
Applications to the extended real numbers, extreme limits.
Algebras and sigma-algebras. Measures.
Outer measures, the Lebesgue outer measure in R^m.
Construction of the Lebesgue measure. Regularity.
Measurable functions, simple functions, a.e.-convergence.
The Lebesgur integral.
Applications. Lebesgue vs. Riemann integrals.

Bibliography	[1] V. Anisiu: Topologie si teoria masurii. Universitatea $Babes-Bolyai$, Cluj-Napoca, 1995.
[2] C. Crăciun : Lecţii de analiză matematică. Universitatea Bucureşti, 1982.
[3] C. Crăciun : Exerciţii şi probleme de analiză matematică. Universitatea Bucureşti 1984.
[4] C. George: Exercises in integration. Springer, New York, 1984 
[5] J. Kelley: General topology. Van Nostrand, Princeton, 1950.
[6] P. Kree: Integration et theorie de la mesure. Une approche geometrique. Ellipses, Paris, 1997
[7] W. Rudin: Real and complex analysis, McGraw Hill, New York, 1988 (exista traducere in limba romana)
[8] G.B. Folland: Real Analysis. Modern Techniques and their applications. Wiley, 1999

Assessment	Midterm test and Final Exam.