Un (r)I g(r). In → 1 nniformly on every compact subinterval of (0,00), and g(x) dx < oo Prove that
Un (r)I g(r). In → 1 nniformly on every compact subinterval of (0,00), and g(x) dx
(12) Suppose that f [0, oo) - [0, o0) and that f E R(0, n), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral So.o)dexists, and f dA f (x)dax lim -- noo 0,00)
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I = [0, 1] and compute the value of f du
(4) Define the function f : R -»R* by x-1/2 r> 0 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I...
4. Define the function f: 0,00) +R by the formula f(x) = dt. +1 Comment: The integrand does not have a closed form anti-derivative, so do not try to answer the following questions by computing an anti-derivative. Use some properties that we learned. (a) (4 points). Prove that f(x) > 0 for all x > 0, hence f: (0,00) + (0,0). (b) (4 points). Prove that f is injective. (c) (6 points). Prove that f: (0,00) (0,00) is not surjective,...
definition of continuity to prove that f : (0,00) by f(x)-13 + 1 is continuous at every Zo 0. Use the є-ð definition ) Use the є- R defined that g(x)-_a_ is continuous at every a є (-1,00) +1
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
(12) Suppose that f: [0, o0) - (0, 00) and that f e R((0, n]), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral Jo.0)f dA exists, and f dA [0,00) lim f (x)dx noo
(4) Define the function f : R -> R* by ,--1/2 f(x) x< 0. +oo, |(a) Prove that f is measurable (with respect to the Lebesgue measurable sets). (b) Prove that f is integrable on I 0, 1and compute the value of = f du
(4) Define the function f : R -> R* by ,--1/2 f(x) x
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f is 0(g)" on F is: (a) (4pt) Write down the definition for "f is O(g)". (b) (4pt) Prove that the relation is reflexive (c) (6pt) Prove that the relation is not symmetric. (d) (6pt) Prove that the relation is transitive.
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f...
XL Xa 12. (a) Suppose that f(x) = g(x) for all x. Prove that lim f(x) < lim g(x), provided that these limits exist. (b) How can the hypotheses be weakened? (c) If f(x) < g(x) for all x, does it necessarily follow that lim f (x) < lim g(x)? Ya X-
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and compute the value of f du
(4) Define the function f : R -> R* by .-1/2 f(x) +oo, (a) Prove that f is measurable (with respect to the Lebesgue measurable sets) (b) Prove that f is integrable on I [0, 1 and...