Suppose that the functions f and g are defined throughout an open interval containing the point x,, that f is differentiable at x, that f(x,)=0, and that g is continuous at x. Then the product function fg is differentiable at x,. Select one: True O False

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

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Q: The following statement is false, explain why. The function g(x) = x^2 on the interval [-1,1] has…

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Q: Assume that a continuous function f:R → R is X- and µ-periodic, where A and u are two nonzero…

A: We know that between two rational number lies infinite many irrational number. But function is…

Q: Let f be continuous from [0, 1] to [0, 1]. By posing 6(x) = f(x) – x, show that fhas a fixed point…

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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: 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: Does there exist a continuous function f(x) such that f(0) all x in (0, 2)? 10, f(2) = 2 and f'(x)…

A: Let's find.

Q: Suppose that the function f: [0, 1] in R is continuous, f(O) > 0, and f(l) = 0.Prove that there…

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

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Q: Let f be the function defined by: sin x, x <0 x2, 0 <x < 1 2 - x, 1<x < 2 x-3, x22 f(x) = %3D For…

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Q: Let Show f is defined for all x &gt; 0. Is f continuous on (0,∞)? How aboutdifferentiable?

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Q: Suppose f is defined on [a, b] and g is defined on [b, c] with f(b) g(b). Then define { f(x) if a <…

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Q: Let 4 – x² if x > 1 f(r) if x < 1 Determine whether or not f (x) is continuous at r = 1.

A: If left hand limit and right hand limit same, then f(x) is continuous at x=1.

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: Suppose that f (x) = x·g(x) for some function g that is continuous at 0. Prove that f is…

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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: Suppose that f is continuous at 0, and that f satisfies f(x + y) = f(x)+ f(y) for all real numbers…

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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: 5) Suppose that f is continuous on (-00, 00). How many non-differentiable points could f have? Is it…

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Q: Show that f is continuous on (∞ , -∞ )  .

A: we have to show that f(X) is continuous in (-∞,∞). f(x)=x2             if   x&lt;1           x…

Q: If a function f is not defined at x = a then it is not continuous at x = a. Select one: O True O…

A: As we know that a function f(x) is said to be continuous at x=a when the right-hand limit and…

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: Let f be a continuous function on R such that f(x+y)=f(x)+(y) ¥x, yER. Show that f(x)3cx for some…

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Q: Suppose f and g are continuous functions on the interval [a, b], then SEAFX) – g(x)) dx = [." f(x)dx…

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Q: Suppose that f and g are continuous on I = [a, b], and they are differentiable on (a, b) with f'(x)…

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Q: Consider the function f with domain (-2, ): 3|x2 – 1| 2x2 + 2x – 4 if – 2< x < 1 f (x) = 3 if x = 1…

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Q: If a function f(x) is not defined at x-a, then it is not continuous at an open interval (a, b) * O…

A: Given Statement is "False"

Q: Suppose f and g are differentiable functions on R such that,

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Q: Let f be differentiable on an interval A. If f'(x) not equal to  0 on A,show that f is one-to-one on…

A: Given that, Let f be differentiable on an interval A. If f'(x) not equal to  0 on A. We have to show…

Q: Define f(2) in a way that extends f(s) = 2 to be continuous at s = 2. %3D -4

A: The function is defined as fs=s3-8s2-4.

Q: Let f(x) = −2/x and g be the function defined b)Is f . g continuous at x = 3? If not, does…

A: limit of the composite function

Q: Is the function f (x) below continuous on [3,+0)? (0, x > 0 X, 0 3 Choose

A: This question is related to limit and continuity, we will solve it using given information.

Q: Suppose f is continuous on [a, b], ƒ'(x) exists at some point x e [a, b], g is defined on an…

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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: and if the function is differentiable at x

A: The Given function is f(x)=x+33  ,  a=-3

Q: Suppose that the function f(x) = {x“ + 1, x>-1 x <-1 is continuous on R, then a = lax, %3D 1 2 No…

A: The function is given by fx = x2+1, x≥-1ax, x&lt;-1 To evaluate: The value of a if the function is…

Q: Suppose that f and g are both continuous functions. Also f, g € [a, b] and that f= fg. Prove that 3c…

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Q: 4. Suppose that f is continuous on a closed interval [a, b] and differentiable on the open interval…

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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: Suppose f(x) is continuous on [0, 1], with f(0) = 2 and f(1) = 0. Then there must be a value x in…

A: Here given that f(x) is continues.

Q: Suppose f is continuous ont he closed interval [a, b). If F(x) is an antiderivative of f(x), for…

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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: For every continuous function f(x) defined on symmetric interval [a,b]. If f(x) is an even function…

A: Here, the given function f(x) is even. So, f(-x)=f(x)

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: Use the definition of continuity to determine whether the shown function is continuous at a = 1.

A: Given:              fx=2-xIf x&lt;11If x=1x2If x&gt;1

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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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.