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

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Problem 1RQ

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4. Suppose f is a real-valued function with domain D and that xo is an accumulation point of D. Assume L₁ and L₂ are limits of fat xo. Use the definition of limit (involving and 8) to prove that L₁ = L₂. 2 1
10. Consider f: (0, 2)→ R defined by f(x) = x". Assume that f has a limit at 0 and find that limit. (Hint: Choose a sequence (x)=1 converging to 0 such that the limit of the sequence {f(x)} is easy to determine.) 1
Let f : [0, 1] → R be a continuous function. Show that the sequence of functions 1 fn(x) (f( х +п -1 + f(- „x + 1 + f(= + )) n is uniformly convergent on [0, 1]. What is its limit function?
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.
3. Assume that f(x) is continuous on [a, b] and x₁,x2,...,n are arbitrary numbers taken from [a, b]. Show that there exists c E (a, b) so that f(c) 1 · [ƒ(x1) + ƒ(x2) + ... + f(xn)]. n == Hint. Use Intermediate and Extreme Value Theorems.
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].