Suppose DCR and f.D R and g:D-R are both uniformly continuous on D. (a) Prove that the function f + g: D-R is uniformly continuous on D. (b) Give a counterexample to show that the function fg: D-R need not be uniformly continuous on D.

Q: If f is continuous on A) True B False on [a,b], then Sº [ f(x a xf(x) dx = x f(x) dx

A: It will solve by part method integration  ∫u v dx = u∫v dx −∫u' (∫v dx) dx u is the function u(x) v…

Q: 34) If f is an even function that is continuous on [-1,3] and if f f(t)dt f₁f(t)dt = 8, then f…

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Q: 2. Recall that we proved f(r) = VI is continuous on its domain [0, 00). Explain why f(r) is…

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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: 4. Use the definition of uniformly continuity to prove that f (x) 1- x is uniformly continuous on…

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Q: Show that if f: A-> B is continuous and A is compavt then f(A) is also compact

A: Let , f : A →B be continuous function. Here , A is compact set. We need to prove that , f(A) is also…

Q: Let f(x) = x sin x. a. Compute f'(x). b. Show that for every M > 0 and every k > 1, there is xo > M…

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

A: Given problem:- If f is not piecewise continuous on [0, ∞),then ℒ{f(t)} will not exist.  True…

Q: If f is continuous on [a, b] and "If(x)| dx = 0, what can you conclude about f ?

A: Solution:Given If f is continuous on a,b and ∫abf(x) dx = 0To conclude about f ?

Q: If f is continuous at a, then f is differentiable at a. O True O False

A: Answer: False.

Q: Suppose f(x) < g(x) and both are continuous on a, oo). If / g(x) dx converges, then so does f(z) dz.

A: Since it is improper integral so we will use the limit concept to prove these fact.

Q: Prove that the function f(x) = 2x is uniformly continuous on R.

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Q: Q2: If f(x) and g(x) are continuous on [a,b]and if f = [g Prove that 3c e[a,b] such that f(c) g(c).

A: From the given information. The functions f(x) and g(x) are continuous functions on [a,b]. The…

Q: Prove that the function f(x) = Root3(x) is uniformly continuous on [−1, ∞). (Hint: consider the sets…

A: Given that fx=3x we have to prove that fx is continuous in interval [-1,∞) lets consider the set…

Q: 3. Use the e, d definition to show that f(x) = x² is uniformly continuous on A = [0,3]. %3D

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Q: Suppose f is a continuous function such that Si f(t)dt = e® In(x) for all x > 0. What is the value…

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Q: Let f be continuous on [a, b] and suppose that f(x) 2 0 for all x e [a,b]. Prove that if L(f)=0,…

A: Given: Let f be continuous on [a,b] and suppose that f(x) is greater than or equal to zero for all x…

Q: Prove that f(x) =√x is uniformly continuous on [0,∞).

A: According to the question, we have to prove that the function fx=x is uniformly continuous on [0,∞).…

Q: Let f : E → R be uniformly continuous. Prove that |f| is uniformly continuous. Is the converse true,…

A: Let f: E → R be uniformly continuous.  Now, we need to show that |f| is uniformly continuous.  Now,…

Q: Let fn (x) uniformly converge to f(x) on [a, b]. If f„(x) E R[a, b], then f(x) E R[a, b] O True…

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Q: Let f and g be continuous funCHons such that So f(x) dx =9,S f(x) dx = 5, and S g(x)dx= -7 what is…

A: Given, ∫06fxdx=9∫36fxdx=5∫30gxdx=-7

Q: Suppose a function f is continuous on [0, 1], except at a = 0.25, and that f(0) = 1 and f(1) = 3. Is…

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Q: If f satisfies a Lipschitz condition on ICR, then f is uniformly continuous on I. a.

A: Since you have sent more than one question, because of our guidelines we can just answer one of…

Q: Assume f(x) is continuous on [a, b]. Prove that if f(x) > 0 for all x € [a, b], then S. f(x) dx > 0.…

A: Extreme Value theorem states that if a real-valued function f is continuous on the closed inteval…

Q: Suppose that f: [0, ) – R is continuous, and that f(x) – l as x → - x. Prove that f is uniformly…

A: See the attachment

Q: Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b] and if…

A: It is given that,  f(x) is Riemann-Stieltjes integrable function with respect to g(x) on a, b such…

Q: Use the definition to prove f =1 is uniformly continuous on [0, 0).

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Q: If f(x) is differentiable in (a, b) and there is M > 0 such that |f'(x)| < M for all x E (a, b),…

A: Topic = Functions True

Q: Ler f : R → R be continuous. If f(c) > 0, prove there exists a neighborhood N of c such that f(N) >…

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Q: Use the definition of uniform continuity (i.e., the e, d definition) to prove that f(x) = x² – x is…

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Q: 1. Show (in terms of  ε - δ) that a function f : R3 → R defined by f(x; y; z) = (2x + 3y + 4z) is…

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Q: Determine whether the statement is true or false. If f is continuous on [0, o0) and f(x) dx is…

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Q: 2. Show that the function f(x) = cos x is uniformly continuous on

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Q: Let T: (X,d) (Y,d) be a continuous function for all n. If {T,} converges uniformly to T on X, then T…

A: Given Tn:X,d→Y,d^ is a continuous function for all n and Tn converges uniformly to T on X. We have…

Q: (a) Prove that if m ₹ N, then the function f(x) = xm is continuous on R.

A: We have to prove that, the function f(x) = xm,   m∈ℕ  is continuous on ℝ.

Q: If f(x) is continuous on a, b] thenF(x) = So f(t)dt is continuous. OTrue False

A: We need to check whether the given statement is true or false

Q: Use the definition to prove that the function in f(x) = 2x - 3 is uniformly continuous.

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Q: If f is continuous on [a, b] and cE[a, b], then f(x)dx does not exist. Select one: True False

A: Topic:- definite integration of continuous function

Q: Let D := [0,1] and let ƒ : D → R be the function defined by f(x) = Vr. Show that f is uniformly…

A: Consider the provided question,

Q: If |fn(x) – fn(y)l 0 and all x, y in a compact interval, show that {fn} is uniformly…

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Q: If f is integrable on [a,b], show by example that F (t) = | f (x) dx is continuous but need not to…

A: We need to show by an example that if f is integrable on [a, b], then F(t) (as defined in the…

Q: Suppose that f is continuous on [a.b] that f(x) 20 for all xE[a,b] and that f =0 1. Prove that f(x)…

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

Q: (a) Let f be differentiable on (a, b) and suppose that f'(t)| 0 and for all t e (a, b). Prove that…

A: We solve here 3(a), In question given two problem are separated problem. According to our guideline,…

Q: Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] and on [1,∞). Show that it is…

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Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

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Q: Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b]and if…

A: detailed explanation mentioned below.

Q: 1. Given that f is continuous on a, b|, compare f(b)(b – a) | f(x) dx, and if f is constant on a,…

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Q: Suppose f is continuous and F'(x) = f(x). J s(x)dx =3, f(x)dx = 4, [f(x)dx= 5, and F(1) =6, evaluate…

A: Here this is simple application of integration and differentiation please watch carefully.

Q: 1. Use the e-ổ definition to prove that f(x) = x² is uniformly continuous on [a, 이.

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.