390 INDEX

connected, 111inverse Fourier transform, 294inverse image, 46, 56iterated integrals, 216

Jordan condition, 238jump discontinuity, 133

Kurzweil, 303

L’Hopital’s rule, 172L’Hopitals rule, 174Laplace transform

obvious properties, 242unique, 298

Laplace transformsjustification of method, 246

Laplace transorminverse, 298

least common multiple, 26least upper bound, 27Lebesgue

fundamental theorem of calculus, 354Lebesgue integral

desires to be linear, 340nonnegative function, 337simple function, 338

lim inf, 78properties, 79

lim sup, 78properties, 79

limit comparison test, 89limit of a sequence

derivative, 152limit of a subsequence, 63limit of nth root of n, 157limit point, 69limit points, 110

closed sets, 70limits

continuity, 131cosine and sine, 163existence of limits, 79properties, 130squeezing theorem, 63, 130uniqueness, 61, 129well defined, 129

limits of sequencespreservation of order, 63

properties, 62Lindemann Weierstrass theorem, 373Lindemannn Weierstrass theorem, 366Liouville’s theorem, 274Lipschitz, 201Lipschitz constant, 201little o notation, 134ln, 166

derivative, properties, 166local extrema, 142local extremum

derivative equals 0, 142local maximum, 142local minimum, 142locally finite, 75logarithm

analytic, 263branch, 263branches, 262principal branch, 262, 263

logarithmsother bases, 167properties, 167

lower semicontinuous, 115

mathematical induction, 19max. min.theorem, 108maximal chain, 50maximal function

Hardy Littlewood, 346mean square convergence, 287mean value theorem

Cauchy, 146for integrals, 222integrals, 215second, 239usual version, 146

measurabilitylimit of simple functions, 334

measurable function, 333measurable functions

continuous composition, 335measurable sets, 325measurable space, 323measure, 323

properties, 324measure of intervals, 321measure space, 323