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Let SCR and f:S →R be a function. Then f is measurable function if and only if for every open interval (a, b) in R, f-'(a, b) is measurable set.
Let I ⊆R be an open interval,let c ∈I,let f,g:I→R be functions and let k ∈R.Suppose that f and g are differentiable at c. f-g is differntiable at c and |f-g|' (c)=f'(c)-g'(c)
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function which is integrable on a set A E A. Let f, : O -> R* be a sequence of measurable functions such that g(x) < fn(x) < fn+1(x), for all E A and n E N. Prove that lim fn d lim fn du noo A
(11) Let (,A. /) be a measure space. Let g 2 - R* be a measurable function...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1) Show that the inverse function f -1 exists. (2) Prove that f is an open map (in the relative topology on I) (3) Prove that f1 is continuous
Let R be an interval (open, closed, neither are all fine) and let f: I-> R be a continuous strictly increasing function. Do the following: (1)...
4. Let S be the set of continuous function f: [0;1) ! R. Let R be the relation defined on S by (f; g) 2 Rif(x) is O(g(x)). (a) Is R reflexive? (b) Is R antisymmetric? (c) is R symmetric? (d) is R transitive? Explain your answer in details. Use the definition of big-O to justify your answer if you think R has a certain property or give a counter example if you think R does not have a certain...
Let f be a twice differentiable function on an open interval (a, b). Which statements regarding the second derivative and concavity are true? If f"(c) is positive, then the graph of f has a local maximum at x = c. The concavity of a graph changes at an inflection point. If f is increasing, then the graph of f is concave down. The graph of f has a local minimum at x = c if f"(c) = 0. The graph of f is concave up if...
Recall that (a,b)⊆R means an open interval on the real number line: (a,b)={x∈R|a<x<b}. Let ≤ be the usual “less than or equal to” total order on the set A=(−2,0)∪(0,2). Consider the subset B={−1/n | n∈N,n≥1}⊆A. Determine an upper bound for B in A.. Then formally prove that B has no least upper bound in A by arguing that every element of A fails the criteria in the definition of least upper bound. Note: make sure you are addressing the technical...
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove that for any A E A f du lim fn du A 4 (You must show that the integrals exist.)
(3) Let (2,A, /i) be a measure space. Let f : N > R* be a nonnegative measurable function. Define the sequence fn(x) = min{f(x), n}, n E N. Prove...
(4) Let (Q,A) be a measurable space, and let f : Ω-> R. Prove that the following statements are equivalent: f is measurable . f-(I) E A for any open interval I CR .f-(A) E A for any open set ACR. . f-(A) E A for any Borel set ACR.
(4) Let (Q,A) be a measurable space, and let f : Ω-> R. Prove that the following statements are equivalent: f is measurable . f-(I) E A for any open...