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6*. Let f : [O, 1] R be a continuous integrable function an...

6*. Let f : [O, 1] R be a continuous integrable function and suppose that 1 1. X nf(x)dx=0 for any n € N. Show that f must equal to zero everywhere on [O, 1]. 7. Verify that the uniform limit of 80 fn(x) = 4-n n=1 on [O, , 1] is a continuous function which is not differentiable at an...

If (a) Prove that logn &lt; n1/8 for all sufficient...

If (a) Prove that logn &lt; n1/8 for all sufficiently large values of n. (b) Explain why part (a) shows that the series 1/(logn) diverges. (a) Show that if E an is a convergent positive-term series, then the series E sin(an) also converges. (b) Prove that if E an is a convergent positiv...

(1) If f : [0, 1] R is continuous a.e., then f is measurabl...

(1) If f : [0, 1] R is continuous a.e., then f is measurable. (2) If f : E R is continuous except at finitely many points in E, then f is measurable. (3) If {fn} is a sequence of measurable functions E, then the set {x E E: lim fn n(x) exists} n 8 is measurable. (4) There exists a non-measur...

1. Let C C Lo, 1] be the Cantor set and O: [o, 1] TR be th...

1. Let C C Lo, 1] be the Cantor set and O: [o, 1] TR be the Cantoe function (which, as you know, IS monotone increasing). Prove that at no point xcC is 0 differentiable. (So the set of points where a monotone function fails to be differentiable may well be uncountable. ) 2. Consider the Dirich...

EXERCISE 1. Let (ak)kez be a family of numbers indexed by Z....

EXERCISE 1. Let (ak)kez be a family of numbers indexed by Z. We say that Ekez ak C-converges if and only if the series E'bn converges, where bn = an + a-n for all n € N. 1. Prove that Ekez ak C-converges if and only if the sequence (Cr--n ak)nen converges. 2. Assume that the series Enzo an a...

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