# 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.