(Rudin, Ch. 6, Exercise 2) Suppose f > 0, ƒ is continuous on [a, b], and | f (x) dx = 0. a Prove that f(x) = 0 for all E [a, b].

Q: Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove that…

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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: 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: Find all values of c such that f is continuous on (-0, o0). S1-x², {. f(r) %3D x, C = (smaller…

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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: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

Q: 2. Show that f(r) = sin r is uniformly continuous on (0, 00). %3D

A: 2. We need to prove that sinx is uniformly continuous on 0,∞. Definition of uniformly continuous…

Q: Theorem 7. Let X be a continuous RV, g a function from R to R. Let Y = g(X). Then E[Y] = E[g(X)] = /…

A: The expected value of random variable is the weighted average of all possible values of the…

Q: Suppose that f(x) is a continuous function on [a , b] and F(x)= | f(t) dt a Part 1 of the…

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Q: Theorem 11. Let f be a continuous function on R such that f(x+y)-Df(x)+(y) vx, yER. Show that…

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Q: 3. Use the definition to show that the function f(x) = is continuous on the interval [c, o0), where…

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Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

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Q: The function f is a composite of two continuous functions: y = cos x and y = x 2, so f is continuous…

A: The function fx is a composition of functions y=cosx and y=x2and it is a continuous function.

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: Use Theorem 9.2.1 to show that T is not continuous at 0, no matter what value is chosen for b.

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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: Theorem 11. Let f be a continuous function on R such that f(x+y)Df(x)+fy) vx, yER. Show that…

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Q: c. Show that f(x) is NOT uniformly continuous on [0, +∞).

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Q: (a) Assume that g(x, y) is a function of the form g(x, y) = {" 0 if (x, y) = (0,0). (i) Show that…

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Q: Prove thait if y=f(x) f(x)=0 when x is a rotional number, then f(x)=0 for all x € R. is continuous…

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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 f and g are continuous functions on [a,b] and that g(x) > 0 for all x E [a, b]. Prove that…

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Q: f : R → R be defined by f(x) = x² + x. Pro nition, that f(x) is continuous at all a E R.

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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: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

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Q: Prove that if f is continuous on [a, b] and F(x) = ſx ƒ (t)dt α then F is continuously…

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

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Q: Suppose f: a,b] → R is continuous and faf=/ for all xie [a,bl Show that f(x)= 0 ft rEla,b).

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Q: Let f be a continuous (and therefore uniformly continuous) function on [a,b). Show that, given e> 0,…

A: Consider the given information.

Q: Let f, g : D → R be continuous at c E D. Prove that fg is continuous at c.

A: Let ε<1 and M= maxf(a), g(a), 1, then there exists δ>0 such that f(x)-f(a)<ε3M (since f is…

Q: Define f(0,0) in a way that extends f to be continuous at the origin. 6x² -x²y² + 6y² x² +y? f(x,y)…

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Q: Use the definition to prove f = (x-1)/(x+1) is uniformly continuous on [0,∞).

A: Given function is fx=x−1x+1. We have to show that fx=x−1x+1 is uniformly continuous on 0,∞. Let us…

Q: Use the definition of continuity and limit (the "e – 6" definition) to show that the function…

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Q: Theorem 12. Let f be a non-zero continuous function on R uch that f(x+y)=f(x) f(y) ¥x, yER. Show…

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Q: Show that the function f(x) = csc(x) is not uniformly continuous on (0, pi/2]

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Q: Let f (x) be a continuous function on R-{0} such that a a f (x) dx exists for all a > 0.If g(x) = [?…

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Q: If f is continuous on [a,b], then the function g defined by g (x) = S f (t) dt, a < x < b is…

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Q: If fCxl is continuous on [2,7],then Si f(u) dvz f (7)-(2) True e or false?

A: we have given that f(x) is continuous on[2,7].

Q: Prove the following result: A function f that is decreasing on [a,b] is integrable on [a,b].

A: Prove the result that,        "A function f that is decreasing on[a,b] is integrable on[a,b]''

Q: alue of c that makes the following function continuous: S 1+ cx if x 2.

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Q: 4. 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: Suppose f is a continuous on R and f(x + y) = f(x) + f(y) for all r,y R. Prove that for some…

A: What is Continuous Function: In mathematics, a continuous function is one that, in response to a…

Q: 3. Let f, g : D →R be continuous at c € D. Prove that fg is continuous at c.

A: Since you have asked multiple questions, we have answered the first question. If you want the second…

Q: Let f: (0,0) → (0, 0) be a function defined by f (x) =- for all x E (0, ∞). Prove that fis…

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Q: f, g: A → R be continuous on A, and let h(x) Let = max{ f(x), g(x) } for x belong to A. (a) Show…

A: Given, f, g: A→R be continuous on AAndhx=maxfx, gx for x belong to A.

Q: 2. Let f (x) = cos-. Show that f is NOT uniformly continuous on (0,1).

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Q: Suppose f ≤ 0 , f is continuous on [a, b] and ∫ab f dx= 0. Prove f(x) = 0, ∀ x ∈ [a, b].

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Q: 1. Suppose f(r) is a continuous function on r € [a, b]. Prove f(x)dr < S°S(x)| dx.

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Q: Suppose that a function y = f(x) is continuous on I = [a,b] and if the set f(I) = [f(a),f(b)], then…

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