Limits of functions and topology of the real line
- Infinity as the value of the limit: .
- Functions of real variable: the domain, range etc (recall). Examples: polynomial and rational functions, . Compositions: .
- 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: , the same with . Warning: . Operations on infinitely many sets (unions, intersections).
- Local and non-local properties of functions.
- Continuity via open/closed sets.
- Compactness and its implications.