Answered: 1.1. For r € (0, 1] (1) = f(r), wher

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1.1. For r € (0, 1] (1) = f(r), wher

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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Question

In Example 2.4.1, show that fn(x) converges to f(x) pointwise on [0, 1]. (We have
seen the convergence at x = 0, 1.)

For x ∈ [0, 1] and positive integer n, let fn(x) = xn.
Then limn→∞ fn(x) = f(x),, where
f(x) = 0 if 0 ≤ x < 1,
1 if x = 1.
This example shows that the pointwise limit of a sequence of continuous functions need not be continuous.

Question 5. In Example 2.4.1, show that fn(r) converges to f(x) pointwise on [0, 1]. (We have
seen the convergence at r = 0, 1.)

Example 2.4.1. For rE 0,1] and positive integer n, let f,(r) = z".
Then lim fa(r) = f(r),, where
0if0<r<1,
1 if z = 1.
f(2)
This example shows that the pointwise limit of a sequence of contin-
uous functions need not be continuous.

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