2) Let f be continuous on [a, b], differentiable on (a, b) and positive (i.e., > 0) for all x E (a, b). Prove that there S'(c) exists c E (a, b) such that- f(c) + r: (Hint: consider the function F (x) = (x – a) (x – b) ƒ (x) and use b-c MVT for F (x) to show the existence of such c E (a, b).)

Q: 1. Show that the function, defined for x E R by f(x) = 1 is uniformly continuous 1+x² on R.

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Q: consider the function f defined on [0,∞),f(x)=(x^r)sin(1/x),for x≠0 and f(x)=0 and for x=0, where r…

A: Solve the following

Q: Define f : R → R by f(x) = 0 if x∈ R\Q f(x)=1/q if x=p/q with (p, q) = 1. Prove that f is…

A: Define f:ℝ→ℝ byf(x)=0if x∈ℝ\ℚ1qif x=pq with (p,q)=1Prove that f is continuous on ℝ\ℚ and…

Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

A: Solution is given below:

Q: 5. Let f : R → R be defined by f(x) = { 0 if r € Q, x2 otherwise. Prove that f is continuous at 0…

A: Here we prove the problem using the definition of continuity and a theorem.

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: Theorem 11. Let f be a continuous function on R such that f(x+y)-f(x)+fv) vx, yER. Show that f(x)3cx…

A:

Q: 1. Let the function f: R R be given by 1- cos(x) if x + 0, f(x) := if x = 0. 1 Prove that f is…

A: Differentiablity analysis problem....

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

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Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,... then show that f(x)=0…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) | f(x, y) : x2 + Y (x, y) + (1, –1) x + y is…

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Q: Px (w) is a continuous function of w in (- ∞, 0) for any real value of ow, and in fact, it is…

A:

Q: Theorem 11. Let f be a continuous function on R such that f(x+y)3Df(x)+f) x, yER. Show that f(x)3Dcx…

A: Since you have posted a multiple questions. So,  we will solve only first question  for you. To get…

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: Suppose f is differentiable on some open interval containing e and f" (c) = 0 then (c, f(c)) is a…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + Y (x, y) # (1, –1) x +y is…

A: Here, the given function is: f(x,y)=1;   (x,y)=(1,-1)x2+yx+y; (x,y)≠(1,-1) lim(x,y)→(1,-1)

Q: Define f(0,0) in a way that extends f to be continuous at the origin. 2.2 11x - xy + 11y 2 f(x,y) =…

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Q: Prove that if f (x) is continuous at x = a and g(y) is continuous at y = b, then F(x,y) = f(x)g(y)…

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Q: If f is continuous everywhere except at x = 2, then f(x +1) dx is not integrable. True False

A: To check whether the given statement is True or False."If f is continuous everywhere except at x=2,…

Q: 2. Show that the function, defined by f(x) = is NOT uniformly continuous on its x2 %3D domain D(f) =…

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Q: Show that f(x) = In x is not uniformly continuous on (0,2)

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Q: (b). Show that iff= u+iv is analytic in a region S and u is a constant function (i.e., independent…

A:

Q: 0. If y = Ax) is continuous at x = xo, and z = yo =Axo), prove that z = g[f{x)] is continuous at…

A: Explanation of the answer is as follows

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 : V → W is differentiable at a point a ∈ V it must be continuous there.

A: I have proved limiting value is equal to functional value

Q: 9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) %3D f(x, y) = { a2 + y (x, y) # (1, –1) x +…

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Q: Prove that the function f(r) = is uniformly continuous on (0, 00) エ十2

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Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,.. then show that f(x)3D0…

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Q: Let the function f: R R be differentiable at x = 0. Prove that lim %3D = 0.

A: Given the function f:R→R be differentiable function at x=0. We have to prove that limx→0fx2−f0x=0.…

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

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Q: Suppose that f : R →R is continuous and that its image f(R) is bounded. Prove that there is a…

A:

Q: If f is a continuous function on [a, b], ƒ is differentiable on (a, b), and if both f(a) and f(b)…

A: This question is based on Rolle's theorem which is explained in step 2.

Q: 2. Prove that if f: I R is continuous on I, then |f|: I – R where |F(x) := |f(x)| Vz E I is…

A: f is defined on the set of real numbers. The detailed solution is as follows below:

Q: Show that the converses of parts (a) and (b) of Theorem 6.2.8 are false by finding counterexamples.…

A: Theorem: Let f be a differentiable function on the interval I. Then, If f'x>0 for all x∈I, then…

Q: Show that a function f : R2-R defined by f(x,y)= x3- y3 / x2+ y2, if (x,y) is not equal to (0,0)…

A: What is Continuous Function: A function is referred to be continuous at a certain point in its…

Q: Find an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) → 0…

A: To give an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) →…

Q: 15. Let f, g : R → R be continuous at a point c, and let h(x) = sup (f (x). 8(x)} for x e R. Show…

A: A function is one-one means if the value of the domain are equal then the value of the range set…

Q: prove that if f(x)is derivable at a point say "a" then it is continous at x=a

A: Here we have given that the function fx is diffrentiable at a point  .x=a that is f'x exists.…

Q: 5.Prove that: if f is continuous at x = 0, then xf (x) is differentiable at x = 0.

A: We want to prove the given result.

Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

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Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

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Q: (2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies…

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Q: Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = { x2 + y (x, y) # (1, –1) x + y is…

A: Given function is fx,y=1,           x,y=1,−1x2+yx+y  ,x,y≠1,−1 . We have to show that the given…

Q: Show that f : (0,∞) → R defined by f(x) := 1/x is not Lipschitz continuous.

A: Definition:  Let S⊂ℝ and a  function f:S→ℝ is Lipschitz Continuous at x∈S if there is a constant C…

Q: 1. Estimate the limit of the signum function s(x) as x approaches zero from the right. (-1, x0

A:

Q: 15. Let f. g : R → R be continuous at a point c, and let h(x) := sup (f (x). g(x)} for x e R. Show…

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Q: Suppose that the function f: R → Ris continuous and that its image f (R) is bounded. Prove that…

A:

Q: Σ 1 is continuous function in R. Prove f(x) = 4 n=1 n' +x*

A: The given function is                                     f(x)=∑n=1∞1n4+x4 To show that the above…

Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

A: This is a problem of the chain rule and Implicit function theorem.