Real Analysis Answers

{1,-1,2,-2} is a compact set. True or false with full explanation

Let "\\mu" be a finite measure defined on the Borel "\\sigma"-field of R. Prove that

there exists a unique closed set F such that "\\mu"(F)= "\\mu"(R) and such that if

F₁ is any closed set satisfying "\\mu"(F₁) = "\\mu"(R),then F"\\subset" F₁

True or False. Prove if they are true or false or give a counterexample:

1.Let F be a function of bounded variation on [a,b] such that F'(x) = 0 almost everywhere.

Then F is constant.

2.If f is an analytic function the annulus U = {z E C : 1 < Iz| < 4}, then f has an antiderivative in U

If the partition P2 is a refinement of the partition P1 of [a,b], then L(P1,f)≤L(P2,f) and U(P2,f)≤U(P1,f). Verify this result for the function f(x)= 4 cosx , defined over [0, π/2] and for the partition P1= { 0, π/6, π/2} and P2= {0,π/6,π/3,π/2}

If the partition P2 is a refinement of the partition P1 of [a,b], then L(P1,f)≤L(P2,f) and U(P2,f)≤U(P1,f). Verify this result for the function f(x)= 4 cosx , defined over [0, π/2] and for the partition P1= { 0, π/6, π/2} and P2= {0, π/6,π/3,π/2}.

Let the function , defined by

f(x)= 1/x+3 ; x∈ [3,∞[

Check whether f is uniformly continuous or not on the interval of definition

Check whether the interval [7,10[ and ]3,6] are equivalent or not

Show that the set ] -6,8[∩]-8,4[ is a neighborhood of -5.

{1,-1,2,-2} is a compact set. True or false with full explanation