412 INDEX

intervals, 13connected, 115

inverse Fourier transform, 304inverse image, 46, 56iterated integrals, 220

Jordan condition, 243jump discontinuity, 137

Kurzweil, 313

L’Hopital’s rule, 176L’Hopitals rule, 178Laplace transform

obvious properties, 247unique, 308

Laplace transformsjustification of method, 251

Laplace transorminverse, 309

least upper bound, 27Lebesgue

fundamental theorem of calculus, 365Lebesgue integral

desires to be linear, 350nonnegative function, 347simple function, 348

Lebesgue Stieltjes measure, 339lim inf, 78

properties, 80lim sup, 78

properties, 80limit comparison test, 91limit of a sequence

derivative, 156limit of a subsequence, 63limit of nth root of n, 161limit point, 69limit points, 114

closed sets, 70limits

continuity, 135cosine and sine, 167existence of limits, 79properties, 134squeezing theorem, 63, 134uniqueness, 61, 133well defined, 133

limits of sequences

preservation of order, 63properties, 62

Lindemann Weierstrass theorem, 404Lindemannn Weierstrass theorem, 397Liouville’s theorem, 282Lipschitz, 205Lipschitz constant, 205little o notation, 138ln, 170

derivative, properties, 170local extrema, 146local extremum

derivative equals 0, 146local maximum, 146local minimum, 146locally finite, 75logarithm

analytic, 270branch, 270branches, 269principal branch, 269, 270

logarithmsother bases, 171properties, 171

lower semicontinuous, 119

mathematical induction, 19max. min.theorem, 112maximal chain, 50maximal function

Hardy Littlewood, 357mean square convergence, 297mean value theorem

Cauchy, 150for integrals, 226usual version, 150

measurabilitylimit of simple functions, 344

measurable function, 343measurable functions

continuous composition, 345measurable sets, 335measurable space, 333measure, 333

properties, 334measure of intervals, 332, 339measure space, 333measures