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

Code	Subject	
MMA1024	The Role of Counterexamples in Teaching of Mathematical Analysis	

Section	Sem.	Hours: C+S+L	Category	Type	
Didactic Mathematics - in Hungarian	2	2+1+0	speciality	compulsory	
Didactic Mathematics - in Hungarian	4	2+1+0	speciality	compulsory	

Teaching Staff in Charge	Prof. KASSAY Gabor, Ph.D., kassay@math.ubbcluj.ro	
Aims	Systematization and classification of the main concepts of calculus for real functions of one real viariable using different kind of counterexamles. Presentation of wrong solutions in order to avoid them. 	
Content	Course and tutorial 1: Counterexamples relative to sequences.
Course and tutorial 2: Counterexamples relative to series and their convergence criteria. 
Course and tutorial 3: Counterexamples relative to the limit and continuity of real functions. 
Course and tutorial 4: Properties of differentiable functions. 
Course and tutorial 5: Counterexamples relative to the mean-value theorems of differential calculus.
Course and tutorial 6: Counterexamples relative to convex functions.
Course and tutorial 7: Counterexamples relative to graphical representations of functions. 
Course and tutorial 8: Functions with Darboux property.
Course and tutorial 9: Riemann integrable functions. 
Course and tutorial 10: Set with zero measure (Jordan or Lebesgue. Lebesgue criterium for Riemann integrability. 
Course and tutorial 11: Monotone, odd/even, periodics and linear functions. Primitives.
Course and tutorial 12: Mean-value theorems of integral calculus.
Course and tutorial 13: Improper integrals.
Course and tutorial 14: Counterexamples relative to sequences and series of functions.
	
Bibliography	1. Balázs M. -  Hatházi A. :  Matematika,  Erdélyi Tankönyvtanács, Kolozsvár, 2006.
2. Balázs M. :  Matematika analízis,  Erdélyi Tankönyvtanács, Kolozsvár, 2006.
3. Crăciun C.V. :  Analiză matematică (Materiale pentru perfecţionarea profesorilor de liceu), Universitatea din Bucureşti, Facultatea de Matematică, Bucureşti, 1992.
4. Crăciun C.V. : Contraexemple în analiza matematică,  Universitatea din Bucureşti, Facultatea de Matematică, Bucureşti, 1989.
5. Crăciun C.V. : Teoreme de medie din analiza matematică,  Universitatea  din   
Bucureşti,  Facultatea  de  Matematică,  Bucureşti, 1986.
6. Gelbaum B.R. &#8211; Olmsted J.M.H. :  Contraexemple în analiză,  Editura Ştiinţifică,
Bucureşti, 1973.
7. Sireţchi Gh. :  Calculul diferenţial şi integral,  vol. I-II, Editura Ştiinţifică şi   Enciclopedică,  Bucureşti, 1985.
8. Sireţchi Gh. :  Calculul diferenţial, Universitatea din Bucureşti, Facultatea de Matematică,  Bucureşti,  1983.
9. Sireţchi Gh. :  Funcţii cu proprietatea Darboux, Universitatea din Bucureşti, Facultatea  de  Matematică,  Bucureşti,  1986.
10)Rădulescu  S. &#8211; Rădulescu  M. :  Teoreme şi probleme de analiză matemaitică, Editura  Didactică  şi  Pedagogică,  Bucureşti, 1982.
	
Assessment	written exam 40%, homeworks 30%, presentations 30%.