If f : [0, 1] → R is defined as if æ is rational f(x) = 1-x if a is irrational. Ecuss the Riemann Integrability of f(x).

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A: so we have to show that given function is discontinuous at 2 and -2 (a)   Discontinuous  at -2…

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Q: Let f : Z → Z be such that f (x) = x + 1. Is f invertible, and if it is, what is its inverse?

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Q: . Apply Jensen's inequality (2) to f(x) = In x, with i 1 = ..= d = 1/n to prove that 1 Vx1x2...X 0,…

A: Given :-  Apply Jensen's inequality (2) to f  x  = ln x  , with λ1 = λ2 = ..........  = λn = 1n to…

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A: f (x) = ∫0x t2 + a2t4 + a4 atUsing Newton Lebnitt TheoremWe find f'(x) f(x) = ∫m (n)n (n)  g(t) dt∴…

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Q: : Iff (z) is analytic at zo. (a) prove that it must be continuous at zo.

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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…

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Q: - if f(x) is a continuous function at x O, then a = 1-cos4x よく0 x? { a If f(x) = r= 0 r>0 V16+ VF -

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Q: Given that F(x) = a) F'(x) bl 5'(4) 2 1 1 + 2 dt.Using the 2nd Fundamental Theorem of Calculus, find

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Q: Show that the function f(x) = x’ + 1 is invertible from R to R.

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Q: Let f be the function defined on [0, 1] by when x is irrational, f (x) = 1 when x is rational.…

A: Please see the below picture for detailed solution.

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Q: II. Prove the following theorems on limits. 1. f(x) = L if and only if (x) – L] = 0 2. f(x) = L if…

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Q: (1- a? f(x) = 1– x² if x E Q, if r ¢ Q. %3D p at which f is continuous. Prove your conclusion.

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Q: If f is analytic and has a pole at z = a, then f(2)|| → 0 as z → a in any manner.

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Q: yLet f(2) be analytic in {z: |2| < 1} with f(0) that |f"(0)| < 2. = 0 and |f(2)| < 1 for all z in…

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Q: ) Is every rational function on D holomorphic on D?

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Q: If f(z) u * 1V is an analytic function of z then prove that 2B. Əx?

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Q: Let f : (0, ∞) → R be defined by f(x) = ln x. Prove that f is 1-1.

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