Consider the function f : [0, 1] → R, f(x) = x, if x is rational, if x is irrational. For n E Z+, let P₁ be the regular partition of [0, 1] into n intervals. (a) Determine U(f, Pn) and L(f, Pn) explicitly. (b) Evaluate lim U(f, Pn) and lim L(f, Pn). n4x n4x (c) What can you conclude from part (b)? Explain your answer.

Q: 5. Prove that the function f: Z → Z defined by f(x) = 3x + 1 is one-to-one.

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Q: a) f(AUB) = f(A) U ƒ(B). b) (i) f(An B) ≤ f(A)n ƒ(B). (ii) Find a function f {1,2} → {a}, and A, B C…

A: a)  We  have  to prove  that  Now,      Thus,  (Proved)[since, b) i) We  have  to show  that Let  be…

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Q: Use the intermediate value theorem to find the value of c such that f(c) = M. c =

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Q: V

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

A: f: R2 → R, f(x1, x2) = x1-x2 function is given. a) Can f be an surjective function? b) Can f be an…

Q: The range of the function f(x)=2- sin 1x is: X, O a. [-1,3] Ob. [-2,2] Oc. [-3,2] O d. [1,3] O e.…

A: According to question given :f(x)=2-sin-1xWe have to calculate the range of the function.

Q: (b) State the range of f for the domain given in part (a). f:A B f(x)=(x-1)²+4. (c) Define the set B…

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Q: 1. Let b, c, d be real numbers, and let a be a nonzero real number. Consider the function f: R → R…

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Q: Define f : R → R by ƒ(x) = x². Compute the following, or determine if they do not exist. min…

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