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).

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: If f is differentiable at a, then f is continuous at a.

A: If a function fx is differentiable at x=a then function must be continuous at x=a. But a function…

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…

A:

Q: Find an example of a function f:[0,1]¬R such that f is not continuous, but f does satisfy the…

A:

Q: Q2. Let f:R- R be a function such that f(x) = x sin| f is defined at x, , where the value of d…

A: Topic- Limits

Q: 3. Find two real-valued functions f and g that are uniformly continuous on a set E, but such that…

A:

Q: Consider the following example: if g: [a, b] → R is continuous and F is a function defined as F(x) =…

A: It is given that,  g : a, b → ℝ is continuous and F is a function defined as ;  F(x) = 1,x∈a,…

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…

A:

Q: Suppose f" is continuous everywhere. For any constants a and b, show that xf" (x) dx = bf" (b) –…

A: Explained below

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

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Q: For what values of b is - b b + 1' g(x) x? + b, x > 0 continuous at every x?

A:

Q: If f(z) is continuous at z, then it must be differenttiable at z, ylgn O İhi O

A: If a function f(z) is continuous at z0 , then it may or may not be differentiable at z0. Hence, The…

Q: 7. Suppose that f: R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the…

A: Recall the definition of continuity around c .And then choose an ∈&gt;0 to get a δ&gt;0…

Q: 2. Suppose f20, f is continuous on [a, b], and f(x) dx = 0. Prove that f(x) =0 for all x e [a, b).…

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

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Q: Consider the following example: if g: [a, b] → R is continuous and F is a function defined as F(x) =…

A: Please comment if you need any clarification. If you find my answer is useful please put thumbs up.…

Q: If f is continuous on [-1, 1] and f(x)dx = 0, then there exists some ce [-1,1] for which f(c) = 0.

A: Given: f is a continuous function on the interval -1, 1 and ∫-11f(x) dx=0. Prove: There exists some…

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…

A:

Q: If a function f(x) is not defined at x-z, then it is not continuous at interval (a, b) where a > z <…

A: Solution

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: 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: Suppose that f : R → R is continuous at c and that f(c) 0 such that f(x) < 0 in the 8–neighborhood…

A:

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

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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: Prove that f(x) = other than the graphing method. x² – 1 is continuous at c = 0 by any method |

A:

Q: ) For what value of a is fx² -1 x<3 S(x)= { |2ax x23 Continuous at every x?

A:

Q: If f(2) is a continuous function at a point z, , then f(z) is differentiable at z, . But the…

A: Authoring guidelines:…

Q: Q2. Let f:R→ R be a function such that f(x) = x sin (-) has a limit at xo + 0. If f is defined at xo…

A:

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

A:

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: Find all values of a such that f is continuous on R: |x + 1 if x a

A: For a continuous function LHL ( left hand limt) should be equal to RHL (right hand limit)

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: 1 - is continuous at a. f If f(x) 0 for all x in the interval I, and f is continuous at a in I then…

A: 1/f is rational function this is discontinous where either f is discontinous or f is zero  But f is…

Q: If f is differentiable at a ,then f is continuous at a

A: Every differentiable function is continuous.

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: (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: . Is f(x)= Vx + 1 continuous at x == -1? Yes, because all functions are continuous No, because the…

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Q: Suppose a function a increases on [a, b}, a < xo Sb, a is continuous at xo. Let f be a function…

A:

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

A:

Q: Show that the function f:R to R defined by f(x)=5x-12 is continuous using the E-d definition

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

A:

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,…

A:

Q: .For what values of a is Jaax - 2a, x z 2 f(x) 12, x < 2 continuous at every x?

A: Here given function  Since,  it is continuous for every x thus it Is also continuous for x=2

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.