410 INDEX

convergent sequences, 106inverse function, 114limits, 135litany of properties, 108on a compact set, 118one to one, 113preservation of inequality, 108uniform, 118uniform convergence, 121

continuousat one point, 107not differentiable anywhere, 142

continuous and one to onemonotone, 113

continuous function, 106only at irrationals, 110

continuous functionscombinations of, 108supremum and infimum, 120

continuous image of compact set, 112contitional convergence, 96contour integral

definition, 260convergence

of derivatives, 156convergent sequences

continuity, 106convex, 158convex sets

in C, 264cosine and sine

periodic, 168countable, 48countable basis, 73countable intersection of dense open sets, 82counting zeros, 279critical points, 146curve

piecewise smooth, 257

Darboux integrability, 204Darboux theorem, 204decimal expansion of a number, 77Dedekind cuts, 385dense, 21density of rationals, 21derivative

chain rule, 144

complex variable, 257definition, 139equals zero, then function is constant,

150equivalent difference quotient, 144higher order derivatives, 145intermediate value property, 152inverse function, 154mean value theorem, 149product rule, 143quotient rule, 145sum, product, quotient, chain rule, 143well defined, 139

derivative existscontinuous, 140

derivativescosine and sine, 166right and left, 139

derived series, 163diameter of a set, 81differentiable

continuous, 140differential equations

Peano existence theorem, 234differentiation almost everywhere

monotone function, 360differentiation rules, 143Dini condition, 243Dini derivates, 359Dirichlet function, 55Dirichlet integral, 253Dirichlet kernel, 290Dirichlet series, 282Dirichlet test, 95Dirichlet’s estimate, 37discriminant, 31disk of convergence, 162domain, 55dominated convergence theorem, 352double series

absolute convergence, 100interchange order of summation, 100

double suminterchange of order, 100

Dynkin’s lemma, 336

elementary symmetric polynomials, 394entire, 282