n≥ 0 there is a unique holomorphic function h: U → C such that

Q: For a function f: A → B and subsets C and D of A, prove that f(CUD) = f(C) Uƒ(D).

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Q: Define f: [0, 1] →→ R by the property that 1, if is rational, f(x) = a 0, if ar is irrational. True…

A: f:0, 1→ℝfx=1, if x is rational,0, if x is irrational.

Q: Prove: If a > 0, then (1/a) > 0.

A: It is given that a> 0.It is known that if ab > 0 then a > 0 or b > 0 or a< 0 or b…

Q: ts) Let f: Z → Z be given by f(x) = 3x + 1. Show that f is one-to-one. Is f onto? EXPLAIN.

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Q: Prove: If f1, f2 € R(a) on [a, b] with fi < f2, then ' fi da < s' fz da

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Q: Let u = and v =| [:]} Use the Cauchy-Schwarz inequality to show that a² + b? 2 + (*) a +b 2 VI

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Q: Let f : R → R and ECR. Prove f is continuous iff f-¹(E) is closed whenever ECR is closed.

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Q: Let h : C N x N where C CZ given by h(m) = (m2 +1, m2 + 2). Choose all subsets C' such that h is an…

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Q: If f is holomorphic on the D(0, 2) and f(x) = x4 – 2x² for all x = [0, 0.2], show that f(i) = 3.

A: Given that f is holomorphic on the D0,2 and f(x) = x4-2x2 for all x ∈ 0,0.2

Q: 3. Give an example of a function f.N→N which is surjective but not injective.

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Q: Let S = {0,1}. Exhibit all functions S → S and classify them as injective, surjective, bijective, or…

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Q: Let f : R → R be such that for all x, y ∈ R, |f(x) - f(y)| ≤ (x - y)2. Prove that f is a constant…

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Q: Let f(x,y,z,w)=45x3z-1/y+wz3. Evaluate f(a,b,c,d) where a,b,c are non-zerol real numbers

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