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

Code	Subject	
MMA0013	Supplement of Mathematical Analysis	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics - in Romanian	6	2+1+0	speciality	optional	
Mathematics-Computer Science - in Romanian	6	2+1+0	speciality	optional	

Teaching Staff in Charge	Prof. DUCA Dorel, Ph.D., dduca@math.ubbcluj.ro
Assoc.Prof. FINTA Zoltan, Ph.D., fzoltan@math.ubbcluj.ro	
Aims	Presentation of the main complementary notions and results in Mathematical Analysis.	
Content	1.	Real numbers. Complements.
2.	Sequences: upper and lower limits.
3.	Upper and lower limits of functions.
4.	Upper and lower semicontinuity.
5.	Derivatives. Dini&#8217;s numbers.
6.	Lagrange&#8217;s theorem: the determination of the position of an intermediate point, contraction intervals.
7.	Lagrange&#8217;s theorem: properties of intermediate points.
8.	Cauchy&#8217;s theorem: properties of intermediate points.
9.	Taylor&#8217;s theorem: properties of intermediate points.
10.	Mean theorems of integrals: properties of intermediate points.
11.	Henstock-Kurzweil&#8217;s integral: definition, examples, characterization.
12.	Properties of the Henstock-Kurzweil integral.
13.	The connection between Henstock-Kurzweil integral and the Riemann integral.
14.	The characterization of the antiderivative functions with the help of strong Henstock-Kurzweil integrable function.
	
Bibliography	1.	D.I. Duca, E. Duca: Exercitii si probleme de analiza matematica, Editura Casa Cartii de Stiinta, Cluj-Napoca, 2007 (vol. 1), 2009 (vol. 2)
2.	S. Leader: The Kurzweil-Henstock integral and its differentials: a unified theory of integration on R and R^n, Marcel Dekker, Inc., Basel, 2001
3.	M. Megan: Bazele Analizei matematice, vol. 1,2,3, Editura Eurobit, 1997, 1997, 1998
4.	A. Precupanu: Analiza matematica (Functii reale), Editura Didactica si Pedagogica, Bucuresti, 1976
5.	Gh. Siretchi: Calcul diferential si integral, vol. I si II, Editura Stiintifica si Enciclopedica, Bucuresti,1985
6.	Gh. Siretchi Gh.: Functii cu proprietatea Darboux, Universitatea din Bucuresti, Bucuresti, 1986
7.	V.A. Zorich: Mathematical Analysis, Springer, Berlin, 
	
Assessment	Exam.The activity ends with a written final exam (50%). During the semester, the students will have to prepare two reports (25%).  The students&#8217; activity during the semester will be also considered (25%).  
All university official rules with respect to students&#8217; attendance of academic activities, as well as to cheating and plagiarism, are valid and enforced.