converse, 593
cosine function, 233 , 313, 431, 538-
540
countable set, 126
Darboux
integrable function, 364
lower and upper integral, 363
lower and upper sums, 361
de Morgan's laws
in logic, 591
in set theory, 616, 617
decimal representation, 91
base-b, 165
base-three (ternary) , 166
base-two (binary), 168
nonterminating, 166
terminating, 166
deleted neighborhood, 188
dense
d. in itself, 34
d. set in an ordered field, 33
denseness
of rationals in R, 33, 76
of irrationals in R, 33, 76
of one set in another, 151
denumerable set, 126
derivative, 297 , 300
chain rule, 309
from the left/right, 302
intermediate value property, 319
notation for, 306
of a composite function, 309
of a constant multiple off, 306
of a monotone function, 318
of a power function, 305, 311-
313
of a product, 306
of a quotient, 306
of a sum or difference, 306
of absolute value, 298
of Cantor's function, 321
of exponential functions, 312,
313, 426, 427
Index 729of logarithm functions, 312
of the sine function, 430, 431
of trigonometric functions, 313,
432
power rule, 305, 311-313
derived series, 509
diameter of a set, 163
differentiability implies continuity,
300
differentiable from left/right, 302
differentiable function, 297, 300
differentials, 409
Dini's uniform convergence theorem,
565
Dirichlet function, 187, 228, 240, 364,
393, 450
Dirichlet's test for series, 495, 554
discontinuities of a function
sets of, 290 , 292, 295
types of
essential, 239
infinite, 240
jump, 239
oscillating, 240
removable, 238
simple, 239
discontinuous function, 226
disjunction, 585
distributive laws
in logic, 593
in set theory, 616, 617
distributive property, 3
divergence
of a sequence of real numbers,
52
of a series of real numbers, 453
of improper integrals, 434, 435,
438, 439
to infinity, 82, 100
division, 8
domain of a function, 619
dominates, 463
dot product, 499