388 INDEX

inverse function, 110limits, 131litany of properties, 104on a compact set, 114one to one, 109preservation of inequality, 104uniform, 114uniform convergence, 117

continuousat one point, 103not differentiable anywhere, 138

continuous and one to onemonotone, 109

continuous function, 102only at irrationals, 106

continuous functionscombinations of, 104supremum and infimum, 116

continuous image of compact set, 108contitional convergence, 93contour integral, 253convergence

of derivatives, 152convergent sequences

continuity, 102convex, 154convex sets

in C, 256cosine and sine

periodic, 164countable, 48countable basis, 73countable intersection of dense open sets, 81counting zeros, 271critical points, 142curve

piecewise smooth, 251

Darboux integrability, 200Darboux theorem, 200decimal expansion of a number, 77Dedekind cuts, 357dense, 21density of rationals, 21derivative

chain rule, 140complex variable, 251definition, 135

equals zero, then function is constant,146

equivalent difference quotient, 140higher order derivatives, 141intermediate value property, 148inverse function, 150mean value theorem, 145product rule, 139quotient rule, 141sum, product, quotient, chain rule, 139well defined, 135

derivative existscontinuous, 136

derivativescosine and sine, 162right and left, 135

derived series, 159diameter of a set, 80differentiable

continuous, 136differential equations

Peano existence theorem, 230differentiation almost everywhere

monotone function, 349differentiation rules, 139Dini condition, 238Dini derivates, 348Dirichlet function, 55Dirichlet integral, 247Dirichlet kernel, 280Dirichlet series, 274Dirichlet test, 92Dirichlet’s estimate, 37discriminant, 31disk of convergence, 158domain, 55dominated convergence theorem, 341double series

absolute convergence, 97interchange order of summation, 97

double suminterchange of order, 97

Dynkin’s lemma, 326

elementary symmetric polynomials, 363entire, 274epigraph, 116equicontinuous, 122