7. Prove that C[−1,1], i.e. the set of continuous functions on [-1,1], with thedistanced(f,g) = | |ƒ(x) – g(x)|dx,is a metric space that is not complete. That is, find a Cauchy sequence in C[−1, 1]that does not converge to an element in C[−1,1].

We have to prove that the set of continuous function on -1,1 with df,g=∫-11fx-gxdx is a metric that…

Answered: 7. Prove that C[−1,1], i.e. the set of…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Prove that the function |2x - f(x) = 32x3 6-4x does not tend to a limit as x tends to 3. Use the e- definition of continuity to prove that the function f(x) = x3 + x²-4x is continuous at the point c = 2. Prove that the function f(x) = x-2 2x+1 is uniformly continuous on the interval [1, 3], referring to any results from the module that you use.
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Suppose f: (0, ∞) → (0, ∞) is a continuous function, and g: (0, ∞) → (0,00) is defined by the property that g(x) = (f(x))² for every positive real number . True or false: If g is differentiable, then f must be differentiable too. True False

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7. Prove that C[−1,1], i.e. the set of continuous functions on [−1,1], with the distance d(f.g) = [* |\ƒ(x) — g(x)\dx, is a metric space that is not complete. That is, find a Cauchy sequence in C[−1, 1] that does not converge to an element in C[-1,1].