. Let E be a measurable set with m(F np with ISp <∞. Further if Lim (E) then lI f llo = %3D

Q: Give an example of a function and a point x = a such that the limit of the function as approaches a…

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Q: . Show that the function f defined below from (0, 1] to (0, 1) is (i) well defined (i.e., s every…

A: Injective &amp; surjective function

Q: 3. Show that the functionf on [-1,1] defined as if f (r) = 1+e if x = 0 is discontinuous at x = = 0.…

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Q: 7. If H₁,H2,..., Un are measured in a space X and a₁, a₂,..., an they are real no negatives, then…

A: Given Data:   λE =∑j=1n ajμjE is a measure in X

Q: 6. let (Sa) C Cla, b). Suppose there exsits a function g € Lla, b) with fa(#) S g(x) a.e. for each…

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Q: 5. Find the​ x-value(s) where the graph is not continuous.

A: Step 1: Let write the given.We are given a graph of a certain function. Step 2: Let write what we…

Q: 12. Let f20 and measurable over E. Let et [₁ E f = 0. Prove that f = 0 a.e. Hint: Consider {z € E :…

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Q: 10 8 4 6. 2. 2.

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Q: 6. Let (a, b, c) and let Σ* be the set of all words w using letters from Σ; see Example 2(b). Define…

A: One to one function: A function f is one-to-one if all element of the range of f corresponds to…

Q: Let S (See the notes from Lecture 5 for a picture.) For each of the following, state its value, or…

A: Given: S=A3,A4,A5,... where Ak is the area of a regular k-gon inscribed in a circle of radius 1.

Q: b. lim a-1 ²ln(1-x) X -dx

A: Given

Q: s let X and Y be two top. S paces ancd f:X-y be a function: If f ý arbitrary, x discrete , Y…

A: let X and Y be two topological spaces  Y be any arbitrary topological space. and f:X→Y be any…

Q: 7. Show that lim x³ = c³ for any c E R. X-C

A: To prove:- for any

Q: lim, „(Væ² + x – x)=? O a. 0 O b. 1 O c. Limit does not exist O d. 00 O e. 1/2

A: In these type of questions just rationalize the given term and simplify the fraction before finding…

Q: Give an example of functions f and g and a point x = a such that the limit of f as x approaches a…

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Q: In) dx, 1 3 5. a) Evaluate

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Q: 5. Determine if there is a value(s) of c that will make this piecewise function continuous. x+c)2 x…

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Q: Give an example of a function and a point x = a such that the limit of the function as x approaches…

A: We find an example of a function and a point x = a such that the limit of the function as x…

Q: 1. Consider a function : DR where DCR is measurable. Show that the following are equivalent: (i) is…

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Q: Show that for all x E (0, 00) we have In(x) <x – 1. Also, graph the functions In(x) and a - 1…

A: Suppose f is differentiable on an interval where I can be  . Note the result corresponding to…

Q: 3. Let E be a set of mea E. Suppose that pi > P2 >

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Q: III. 2. Consider the function x<0 0SAS1 (1) At what points is this function continues? (2) For each…

A: fx=1+x2, x&lt;0ex2,      0≤x≤11+x2, x&gt;1

Q: 4. If L is a Continuous function on H, then unique YEH such that L(x)=(x,y). уєн there is a

A: Given that L is continuous on H. To show: y is unique in H.

Q: then A(-2u - 20) = -2 ---B 5 and Au 3 Au = 2 4

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Q: 1. Give an example of a function f : R –→ R that is continuous and bounded but does not achieve…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 2.5. Give an example of a bounded function f and an increasing function a defined on [a, b] such…

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Q: . What is the point discontinuity of the x² – 1 lim x→-1 x +E 1

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