Sergei Yakovenko's blog: on Math and Teaching

Monday, November 16, 2009

Lectures 3-4 (Tue, Nov 17, 24; 9:00-11:00)

Filed under: lecture,Rothschild course "Analysis for high school teachers" — Sergei Yakovenko @ 11:30

Limits of functions and topology of the real line

  • Infinity as the value of the limit: \lim_{n\to\infty} x_n=\pm\infty.
  • Functions of real variable: the domain, range etc (recall). Examples: polynomial and rational functions, \sin x,\ x^\alpha. Compositions: \sin \frac 1x, \ \cos\ln x.
  • Limit of a function: one-sided, two-sided. Continuity points.
  • Sequential limit vs. “standard” limit: equivalence theorem.
  • Open and closed subsets on the real line. סביבות וסגור
  • Images and preimages. Some algebra of sets: f^{-1}(Y\cap Z)=f^{-1}(Y)\cap f^{-1}(Z), the same with \cap. Warning: f(A\cap B)\ne f(A)\cap f(B). Operations on infinitely many sets (unions, intersections).
  • Local and non-local properties of functions.
  • Continuity via open/closed sets.
  • Compactness and its implications.
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