The exam covers the content of (part of) section 11, sections 17, 18, 20. For section 20, the book gives a very general definition applicable to all cases at once; we break it into separate more familiar definitions. - Bolzano-Weierstrass theorem: every bounded sequence has a convergent subsequence. (know the statement, understand what it means. what happens if the sequence is not bounded?) -Continuous functions. Two definitions: via sequences and epsilon-delta. (be able to check whether simple functions are continuous or not, construct basic examples of continuous and discontinuous functions. Understand why two definitions are equivalent) -Operations (sums, products, etc) with continuous functions. Continuity of polynomials and rational functions (know proofs). Other examples of continuous functions (not much required here, just know that the basic functions from calculus, sin x, cos x, exp x, etc are continuous. You can use this without proof.) - Properties of continuous functions on closed, bounded intervals: bounded, attains max/min, intermediate value theorem. (know the statements and an outline of proofs. understand where proofs fail if interval not closed/not bounded or function not cont., give counterexamples.) - Limits of functions (two-sided, one-sided, limits at infinity). Sequential and epsilon-delta definition. Theorem: two-sided limit exists <=> both one-sided lims exist and are equal. (understand proof; be able to use the theorem.) Relation between existence of limit and continuity at the point. Various additional properties whose proofs are very similar to those for continuous functions/sequences: limit laws for lim f(x), uniqueness of limit, etc (see practice sheet for more).