By Thomas A. Garrity, Lori Pedersen

ISBN-10: 0521797071

ISBN-13: 9780521797078

Few starting graduate scholars in arithmetic and different quantitative topics own the daunting breadth of mathematical wisdom anticipated of them once they start their reviews. This ebook will supply scholars a large define of crucial arithmetic and should aid to fill within the gaps of their wisdom. the writer explains the elemental issues and some key result of the entire most vital undergraduate themes in arithmetic, emphasizing the intuitions in the back of the topic. the subjects comprise linear algebra, vector calculus, differential and analytical geometry, genuine research, point-set topology, likelihood, complicated research, set idea, algorithms, and extra. An annotated bibliography bargains a advisor to extra examining and to extra rigorous foundations.

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**Example text**

2 A function f : R -+ R is continuous at a if given any E > 0, there is some fJ > 0 such that for all x with 0 < Ix - al < fJ, we have If(x) - f(a)1 < E. For an example, we will write down a function that is dearly not continuous at the origin 0, and use this function to check the reasonableness of the definition. Let f(x) = { 1 ~f x > 0 -11fx::;0 Note that the graph of f(x) has a break in it at the origin. We want to capture this break by showing that lim f(x) x-+o i= f(O). 26 CHAPTER 2. E AND 8 REAL ANALYSIS Now f(O) = -1.

The key concept is that of a limit, from which follow the definitions for differentiation and integration and rigorous proofs of their basic properties. Far from a mere exercise in pedantry, this rigorization actually allowed mathematicians to discover new phenomena. For example, Karl Weierstrass discovered a function that was continuous everywhere but differentiable nowhere. In other words, there is a function with no breaks but with sharp edges at every point. Key to his pl:Oof is the need for limits to be applied to sequences of functions, leading to the idea of uniform convergence.

28 E AND 8 REAL ANALYSIS Part of the power of derivatives (and why they can be taught to high school seniors and first year college students) is that there is a whole calculational machinery to differentiation, allowing us to usually avoid the actual taking of a limit. We now look at an example of a function that does not have a derivative at the origin, namely f(x) = Ixl· This function has a sharp point at the origin and thus no apparent tangent line there. We will show that the definition yields that f(x) = Ixl is indeed not differentiable at x = O.

### All the Mathematics You Missed: But Need to Know for Graduate School by Thomas A. Garrity, Lori Pedersen

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