Continuous Functions of Vector Variables
This text is appropriate for a one-semester course in what is usually called ad vanced calculus of several variables. The focus is on expanding the concept of continuity; specifically, we establish theorems related to extreme and intermediate values, generalizing the important results regarding continuous functions of one real variable. We begin by considering the function f(x, y, ... ) of multiple variables as a function of the single vector variable (x, y, ... ). It turns out that most of the n treatment does not need to be limited to the finite-dimensional spaces R , so we will often place ourselves in an arbitrary vector space equipped with the right tools of measurement. We then proceed much as one does with functions on R. First we give an algebraic and metric structure to the set of vectors. We then define limits, leading to the concept of continuity and to properties of continuous functions. Finally, we enlarge upon some topological concepts that surface along the way. A thorough understanding of single-variable calculus is a fundamental require ment. The student should be familiar with the axioms of the real number system and be able to use them to develop elementary calculus, that is, to define continuous junction, derivative, and integral, and to prove their most important elementary properties. Familiarity with these properties is a must. To help the reader, we provide references for the needed theorems.
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accumulation point arc-connected Assume that f ball Bolzano-Weierstrass theorem boundary bounded magnification bounded sequence bounded set Cantor set Cauchy sequence Cauchy's closed and bounded closed sets closure point compact set Compare Exercise component conclude contained continuous function convex coordinates defined disconnected disjoint distance domain dot product equation equivalent Euclidean space Example exists extreme value property f is continuous Figure finite dimension finite limit finite-dimensional graph Hence implies infinite inner product space interior intermediate value intersection interval isolated point line joining maxnorm metric multiple nonempty nonnegative nonzero normed linear space normed space normed vector space open set orthogonal partition polynomial proj(x Proof Prove Theorem Pythagorean real number says Section 4.3 segment sequentially compact Show Similarly sublimit subspace Suppose Theorem 2.5 triangle inequality unbounded uniformly continuous union vector function vector space zero