9. Consider the function f: R → R, defined by (x²sin(1/x) ifx # 0 f(x) = if x = 0 Then a. f is not continuous. b. f' is not continuous. c. f'(0) does not exist.

Q: (0,1). Prove the following functions is uniformly continuous on sin x f(x) =

A: Uniformly Continuous:-  Here we prove the result using a theorem of uniformly continuous.

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: 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: 5. Show that the function h(r, y) defined by if + y 1 is discontinuous.

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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: Suppose F 1  and F 2 are both antiderivatives of f on an interval I How are  F 1  and F 2  related?

A: Given:  F 1  and F 2 are both antiderivatives of f on an interval I. That is,

Q: If f(x) is continuous at x = c then f'(c) exists. True False

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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: Suppose h' is continuous x>0. IF x = 0 and h'(x) = 3√tan (2x)-1 it exists. i at Find h (0), 'F

A: For the solution of the problem follow the next step.

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: 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: Determine f(5) if f is a continuous function that satisfies +sin(a) f+sin(a) f(t)dt = 22² for all x.…

A: I am going to solve the problem by using some simple calculus to get the required result of the…

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: 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: 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: Clarify that f f :(0,1) →R defined by f(x): not uniformly continuous. =-

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Q: iii. is continuous at a provided g(a) # 0.

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Q: Consider the function xy f(x, y) x²+xy+y² k if (x, y) = (0,0) if (x, y) = (0,0) State, with…

A: Given :  f(x, y) = xyx2+xy+y2if (x, y)≠(0, 0)kif (x, y) = (0, 0) To find : The value of k if it…

Q: 43. Let f(x, y) = Is it possible to define f(0, 0) so x² + y2 * that f will be continuous at (0, 0)?…

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Q: Suppose that fx) is continuous on [0, 9] and F(x) is an anti-derivative of f. If F(0)=9 and F(9)=7,…

A: Given that:                           f(x) is continuous in [0, 9].…

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. Determine if a function f : [0, 2m] →R defined by f(r) = sin 2x is continuous. %3!

A: A metric space is defined when the points in the space follow the following 3 properties:- The…

Q: 11. Prove from the definition that the following functions are differentiable. ()- (b) f =xy?

A: By the definition, the function f(x,y)=xy^2 is differentiable if fx(x,y) and fy(x,y) both exist.

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: that makes y continuous but not differentiable by writing the required procedures to that If: X≥0…

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

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Q: 4. Give an example of a continuous function f : (0, 1) → R that cannot be uniformly approximated by…

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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: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

Q: 2. The following funchion O, x = 0 flad = f(x) |-X, 0<x<l is differentiable on (0,1) and satisfies…

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Q: Let g be a continuous function. Using the substitution u = 3x – 1, the integral (g (3x – 1)) dx is…

A: We can use the concept of method of substitution in order to solve the given problem.

Q: Suppose that f(x) is continuous on [0, 9] and Fx) is an anti-derivative of f. If (0)=9 and F(9)=3,…

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Q: (Read the dėlihition Caleluny.) |x2, if x l; is differentiable at 1. 2x-1, ifx > I; 4. Prove that…

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Q: Determine f(5) if f is a continuous function that satisfies t 4x+sin(rz) f(t)dt = x for all z. %3D

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Q: 2. Consider the function f with domain (-2, 0): 3|22 – 1| if – 2<r<1 2.r2 + 2x – 4 f(x) = 3 if r = 1…

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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: Suppose that f is a continuous function on R and, g and h are differentiable functions on R.…

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Q: Suppose f is continuous on the closed interval [–1,3]. If F is an antiderivative of f, which of the…

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Q: If f(z) is continuous at z then it must be differenttiable at z, 0.

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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: - For what value of b is Sx, bx?, x2 -2 x<-2 х, g(x) = continuous at every x?

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Q: Show that f(x) := x/3, x e R, is not differentiable at x 0.

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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: 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

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Q: Prove f (x) 1+ mx is continuous at x = 1, where m e R

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Q: 2. Suppose that f is a measurable function satisfying m{r ER: |f (x)| > n} ) < 1 nEN. n3 Is f…

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Q: d. Use the limit definition of the derivative, f (x) = lim f (x+h)-f(x) 1 +h)=j, to prove that VT =…

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