Theorem 6.7 (Fatou's Lemma.) Let {fn) be a sequence of non- negative, measurable functions on a set E. If {fn} converges to f a.e. on E, then Lim inf SAx)dx < É Sf.(x)dx n00

Q: 4. (a) Suppose that a function f satisfies |f (x)| < |x| for all x. Show that f is continuous at 0.…

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Q: Define f : Rm \ {0} → R by ƒ(x) = T: Prove that f is uniformly continuous on R" \ B1(0), but not…

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Q: Suppose f: R→R is a continuous, 2n – periodic function. Define the function f,(x) := f(x+ ±) for x E…

A: Given Information -  f is a continuous function on R and is 2π periodic.

Q: Question 3 (a) State and prove sequential criteria J (b) Show by e 8 definition that the function…

A: To Do: (a) State and prove sequential criteria of continuity. (b) Show by ε-δ definition that the…

Q: Prove the given theorem and provide (1) example. Theorem 1.1 If f (x) is continuous in a a <x< b,…

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Q: t if t3 Find a value of A such that the function f(t) is continuous for all t.

A: We have given f(t).  And f(t) is continuous for all t.  It means f(t) must be continuous at t=3

Q: A function f(x) is continuous at an x value of aif and only if the following three conditions are…

A: Concept If a function is continuous at x = a then limx→a-f( x ) = limx→af( x ) = limx→a+f( x )

Q: Give an example of a function that is continuous on -∞ < x < o but is not differentiable at x = 3.

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Q: 3. For each part, sketch a function f that satisfies all the conditions, if possible. If not…

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Q: Use the Cauchy inequalities to show that if f is an entire function that satisfies f(z)| ≤ Czlog(1 +…

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Q: Evaluate , x? dx by limit definition. f(x) dx f(G.) Ax¡ lim i=1

A: Given integral  ∫01x2dx △x=b-an△x=1-0n=1n ci=a+i(△x)ci=0+i1nci= inf(ci)=in2

Q: Let f(x) = { 3, 1< x < 2 0, 2 < a <5 and Let F(x) = { 3x, 1<x < 2 6, 2 <x < 5 Show that wether the…

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Q: True or false: a function f: R → R is continuous at a point a ER if and only if there exists a…

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Q: Answer true or false. If f is not piecewise continuous on [0, ∞o>), then L{f(t)} will not exist.…

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Q: x -9 Check the continuity of f given by f(x)%3= is 0 <x <5 and x 3 at the points 3 %3! x -2x-3 1.5

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Q: Use the e- 8 property in Definition 19.1 to show that f(x) 1 is uniformly continuous on the interval…

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Q: Suppose that f(¤) is a continuous function with f(-5) choice best describes the following statement.…

A: We have to prove given statement .

Q: -5 -5 1 [ f(x) dx = 3, •[ 9(x)dx = 5 6 -6 Let f and g be continuous functions. If

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Q: The graph of f is shown in the figure. 6. 4 х 14 -4 -6+ 2-

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Q: Let x1 O and xn+1 x+1 2 Show that (x,n) converges and find its limit.

A: It is given that  x1=0 and xn+1=xn2+12. Let xn+1=ϕ(xn)=xn2+12 By fixed point iteration method…

Q: Suppose that 0 &lt;ƒ′(x) &lt; 1/2 for all x-values. Show that ƒ(-1) &lt; ƒ(1) &lt; 2 + ƒ(-1)

A: From the given inequality the behaviour of the function can be determined as.

Q: If f"(x) > 0 for all æ, then f is an increasing function. O True O False

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Q: Answer d and e

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Q: Let fn f, where if 0<<! Sn(x) ={ S 2-a if 0<x<1 and f(x) if a = 0 Then lim sup fn (x) – ƒ (x)| =

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