Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. a) Prove that f is continuous at exactly one point, namely at x = 0.

Q: Show there does not exist a continuously differentiable function f on [O, 2] such that f(0) = -1,…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

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

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Q: Suppose that f [a, b] → R is strictly increasing, i.e., that if x₁ < x₂ then f(x₁) < f(x₂). Explain…

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Q: Prove that (r) = tan(r) is uever negative in 0 <r< and show that g(x) = xsinr decreases…

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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: Let f(x) = 9x2, x in QQ = 0, x not in QQ (a) Prove that f is continuous at exactly one point…

A: f(x)=9x2x∈Q0x∉Q

Q: Consider the piecewise function 2+x² x<0 x=0 g(x) = 1, 2-³, 0<x< 1 2-x x 21 (a) For each of the…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

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

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Q: Show that g(x) = (x3-8) / (x-2) has a removable discontinuity at x = 2. How should g(2) be defined…

A: The left-hand limit of a function at a point is the value of the function at a point very close from…

Q: Prove that the following function is continuous at 0. x, if x is rational f(x) = {0, if 0, if x is…

A: Given function is f(x) = x,if x is rational0,if x is irrational

Q: Let 1-tan r f(x)= ExtIE 0,5], 1f f(x) is continuous in [0.), then is 1 -1 /2

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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: Let f(x) = x²+x+ 2 = Z₂ (x) prove that f(x) irreduciable Poly

A: The given function is  f(x)=x2+x+2.

Q: Determine the domain of the function and prove that it is continuous on its domain f(x) = cosx2 /…

A: f(x)=cosx2x2-1

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: f is continuous at a if and only if |f| is continuous at a.

A: False

Q: If f : R+ → R is a function such that f(x + y) = f(x)f(y) for all x,y ∈ R+. Prove that f is…

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Q: Suppose that a function ƒ is continuous on the closed interval [0, 1] and that 0&lt;=ƒ(x)&lt;= 1 for…

A: Let us assume a function g(x)=f(x)-x Since both f and x are continuous on the closed interval [0, 1]…

Q: Let f be a function defined on ]0, 1I[ by : 0, if x is irrational x= p/q; where p and q are positive…

A:

Q: Let f be a continuous function defined on E. Is it true that f-'(A) is always measurable if A is…

A: If you mean AA is Lebesgue measurable, then the conclusion is NO. A classic counterexample is…

Q: ずx 1

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Q: Let f (x) = x2 if x is rational and f (x) = 0 if x is irrational. b) Prove that f is differentiable…

A: Given fx=x2x is rational0x is irrational To prove f is differentiable at exactly one point, namely…

Q: Define f:R → R by ƒ (x) = 5x if x is rational and f (x) = x² + 6 if x is irrational. Prove that f is…

A: Given :  f:ℝ→ℝ  defined by ;         f(x) = 5x,if x is rational.x2+6if x is irrational To prove :…

Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

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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: Prove that f(x) = |x - 6| is continuous at all points c in ℝ.

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Q: What is the continuous extension of f(x) = x3-1 at point x-1. %3D x2-1'

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Q: 2. Determine whether f is continuous or discontinuous at 1. (x – 1 if x > 1 f(x) -x² if x <1 O f is…

A: Given query is to find function is continuous or not at x = 1.

Q: Prove that the following continuous function has at least one real root; h(x) = −2x + 3 − ln x

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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: If a function f(x) is not defined at x-a, then it is not continuous at x-a.

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Q: Consider the function: { 0 ;if x is rational ; if x is irrational f(x) = Use a d, e argument to…

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Q: Let f be a strictly positive, continuous function such that |f(x)| < x for all real x. Does the…

A: if we have 0 ≤ g(x) ≤ h(x) on the interval [a, ∞) then if  ∫a∞h(x) dx  converges  ⇒  ∫a∞g(x) dx…

Q: if x = 0 f (x) = %3D x* sin(-) otherwise

A: In the given question, concept of continuity is applicable. Continuity Continuity for a graph y=fx…

Q: 3-The function f defined by 2. Is f is Continuous at x=| ?

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Q: Prove that f(x) = tan(x) is never negative in 0 < x < ; and show that g(x) = x=l sin x decreases…

A: We have to prove that f(x)=tan(x) is never negative in the interval 0,π2. Then we have to show that…

Q: Determine if g (x) = is continuous at x = 0

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Q: x + 6|, if x - 1 x² – 3x

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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: 45. Let f : R → R be a continuous function, such that f(x)= 0 if x is a rational number. Show that…

A: Given continuous function is f: ℝ → ℝ such that f(x) = 0 if x is a rational number.

Q: Use the definition of continuity to show that h(x) = x^2 + 5x/2x + 1 is continuous at the number a…

A: The given function,   h(x)=x2+5x2x+1

Q: Give a 8, e proof that f(x) = x² + 3x is continuous at x = a, where a is any real %3D number.

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Q: Decide whether or not the real function In (cos 2x) 2x2 x + 0, f(x) = -1, = 0, x = is continuous at…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Give an example of a function f : R → R that is continuous at x if x is an irrational number, but it…

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Q: Use the definition of continuity to determine whether f(x) is continuous at a= 0.

A: Given function, We have to use the definition of continuity to determine whether f(x) is continuous…

Q: Prove that the function f : R → R given by f(x) = 3x° – 2x is continuous at 1 using the a.…

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Q: f (x) = %3D x² +1

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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.