Exercise 3 Define f : Rª\{0} by f(r) = |r|,'. Using the e-5-definition, prove that f is continuous. %3D
Q: interval [a,b] where it is NOT integrable. f(x)= , x∈
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Q: (a) Show that if f is Riemann integrable over [0, 1], then | f(x) dæ = lim / f(x) dx c→0
A: According to Bartleby guidelines; for more than one question asked only the first one is to be…
Q: 6. Let f: R? → R? be a continuous function. Prove that f ([0,1] × [0,1]) is compact.
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Q: Find the linearization L(x) of f(x) at x=a f(x)= x+1/x a=3 show all work
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Q: Exercise 6. Let f be a function with Ifl4 O <1 None of these
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Q: Exercise 9. Let f be any function such that the integral of f over C is 4. Then the integral of - f…
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Q: (d) Is f invertible? If so, what is f-?
A: f:ℝ→ℝ is defined as fx=x1+x⇒fx=x1-x, x<0x1+x, x≥0 f is both 1-1 and onto. Hence, f is…
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Q: (b) Let f be a function defined by 1 if x e Qn [a,b] f(x) = - |-1 if x€ (÷) n [a, b]. Show that fis…
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A: Prove the result that, "A function f that is decreasing on[a,b] is integrable on[a,b]''
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Q: Give a 8, e proof that f(x) = x² + 3x is continuous at x = a, where a is any real %3D number.
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Q: Let A := (0,1] and let f : A → R be defined by f(x) = . Prove that f is continuous on A.
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Q: Every continuous function on R satisfies the conditions of MVT. * False True
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Q: Let f : [a, b] → R be a strictly decreasing function. Prove that f is integrable [a, b). on
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Q: Exercise#5 uniformly continuous on I. Is the converse true? Justify. Prove that if the function f: I…
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Q: Exercise 6. Let f be a function with Ifl<2 and C be the curve that connects 0 to 1 and then 1 to 1+i…
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Q: Exercise 11. Prove that if f: E R is continuous at p E E, then there exists & >0 so that f(x) is…
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- Let (fn) be a sequence of differentiable functions defined on the closed interval [a, b], and assume (fn ) converges uniformly on [a, b]. If thereexists a point x0 ∈ [a, b] where fn(x0) is convergent, then (fn) converges uniformly on [a, b]. Proof. Exercise 6.3.7. Combining the last two results produces a stronger version of Theorem 6.3.1.Let (hn),(tn) be sequences of bounded functions on A that converges uniformly on A to h,t respectively.show that (hntn) converges uniformly on A to htGive an example if f : (0,1) -> R that is bounded, continuous but not uniformly continuous on (0,1).
- Let fn(x) = x^n for x ∈ [0,1]. check if it is pointwise convergence. Define where it becomes discontinuous.Let fn be a bounded sequence of functions uniformly convergetn to f. Prove that f is bounded as well. Is the claim true if we replace the assumption of uniform convergence with pintwise convergence.Determine if the given is divergent or convergent using the theorem.
- Suppose that F(u) denotes the DFT of the sequence of f(x)={1, 2, 3, 4}? What is the value of F(14)? (Hint: DFT periodicity)Let fn be a sequence of functions that are uniformly continuous on a set S. Suppose that (fn) converges uniformly to a function f on S. Prove that f is uniformly continuous on S.Arzela–Ascoli Theorem). For each n ∈ N, let fn be a function defined on [0, 1]. If (fn) is bounded on [0, 1]—that is, there exists an M >0 such that |fn(x)| ≤ M for all n ∈ N and x ∈ [0, 1]—and if the collection of functions (fn) is quicontinuous (Exercise 6.2.14), follow these steps to show that (fn) contains a uniformly convergent subsequence.
- Suppose that a sequence of differentiable functions {fn} converges pointwiseto a function f on an interval [a,b], and the sequence {f′n}converges uniformlyto a function g on [a,b]. Then show that f is differentiable and f′(x) = g(x)on [a,b].Let X be a Banach space an {xn} be a sequence in. X. Provethat if {xn} converges in norm in X, then it converges weakly to thesame limitIn Exercises , determine whether the given integral convergesor diverges. Try to evaluate those that converge