Consider the following example: if g: [a, b] → R is continuous and F is a function defined as F(x) = 1 if x = [a, c] and F(x) = 0 if x € (c, b], the integral gdF exists and is equal to-g(c) Generalizes the above example to the case where F has n discontinuity points at [a, b].

Q: Check whether Rolle's theorem conditions are satisfied or not for the given function f(z) =z- 4z on…

A: The given function is: f(x)=x2-4x Now, the conditions of the Rolle's theorem is: 1) f is continuous…

Q: 7. What is the value of (X-2)2 for the scores 2,3,5?

A: To find : ∑X-22 for the scores 2, 3, 5

Q: Prove the following statement: Assume f is a continuous function on [a, b], k is any real number |…

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Q: f : A → B and g : B → C are functions such that g ◦ f is surjective, then f is surjective. (ii)  If…

A: Introduction: A function f (from set A to B) is surjective if and only if there is at least one x in…

Q: Consider the function f defined by the formula [ Ae*-² + cx- 3c f(x) = x³ + cx² + B if x < 2, if x ≥…

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Q: 2. Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous

A: Determine if a function f : [0 ; 2pie] → R defined by f(x) = sin 2pie is continuous.

Q: : Let f [a, b] → R be a continuous and one-to-one function, where a < b are real numbers. Use the…

A: Introduction: There must be at least one "c" such that c∈(a,b) (or a&lt;c&lt;b) and f(c) = L. if…

Q: Show that the following function is nowhere analytic. f(2) = 2xy + i(x² – y³) %3D

A: Function is said to be analytic at a point if there is a nbd of the point such that f is…

Q: Suppose f: IR→IR and g: functions. True or false: If the composite function f o g is monotonic, then…

A: Given: f:R→R and g:R→R are the continuous function. To Find: To determine whether the following…

Q: Suppose a smooth function f(x) satisfies f(0) = 0, f(1) = 2, and that f[x0,x1,x2] = 1  for any three…

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Q: - Determine whether the following function is analytic f (2) = z Re z + z Im z + z.

A: The objective is to determine whether the given function is analytic or not. Given: fz=zRez+z¯Imz+z¯

Q: . Show that the function f: R→ R defined as f(x) = 13x − 15 is one-to-one.

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Q: Suppose that f and g are discontinuous at x = c. Does it follow that f + g is discontinuous at x =…

A: Let us define that Then both f and g are discontinuous at x=2

Q: If ƒ : R → R and g : R → R are both one-to-one, is f + g one-to-one? If f and g are both onto, is f…

A: As per the guideline I’m supposed to solve only first question from the given questions. And the…

Q: If f : X → Y and g: Y→ Z are functions such that g of is one-to-one, then f is one-to-one.

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Q: f(x) x² - 2x - 8 { *2-22-3 x-4 x = 4 x‡4

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Q: 1. Define function f : Z+ → Z such that f(n) = /n. (a) Is f one-to-one? Justify (prove) your answer.

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Q: Consider the PIECE-WISE function: Ат-В T-2 if r<1, f(x) = { 32, if 1<x < 2, Ba? — А, if a 22.

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Q: Let f: A → B be a function and let a ∈ A and b ∈ B. Does this mean it is true that f(a) = b? If not,…

A: We use definition of function .

Q: Which of the following functions on R is sublinear? a). s1(x) = # x if x ≥ 0 0 if x 0

A: Definition: Let X be a vector space. A function p: X→ℝ is said to be a sublinear function, if it…

Q: d. For each of the following, state whether the functions f: R → R, x → f(x) are injective,…

A: Given functions: fx=3x+2 fx=3x3-7x+3 fx=x6 fx=sinx fx=0.01x fx=4 We have to check whether the…

Q: Prove or disprove: f : Z30 → Z15 given by f([x]) = [x²] is a well-defined function.

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Q: We have shown that the composition of two surjective functions is surjective, i.e. th if f : A → B…

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Q: Suppose f: IR→IR is a monotonic function. True or false: If f is differentiable, then f must be…

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