What is a limit point? Please explain in detail. 38.6. Let f be a continuous function from R to R. Prove that {x : f(x) = 0} is a closed subset of R.Solution. Let y be a limit point of {x : f(x)Yn E {x : f(x) = 0} for all n and limn-o Yn = y. Since f is continuous, by Theorem 40.2 wehave f(y) = lim,→∞ f(Yn) = limn¬ 0 = 0. Hence y E {x : f(x) = 0}, so {x : f(x) = 0} containsall of its limit points and is a closed subset of R.0}. So there is a sequence {yn} such that

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Let f be a continuous function from R to R. Prove that { : f(x) = 0} is a closed subset of R. Solution. Let y be a limit point of {r : f (x) Yn E {x : f(x) = 0} for all n and limn-o Yn = y. Since f is continuous, by Theorem 40.2 we have f(y) = limn- f(yn) = lim,. 0 = 0. Hence y E {r : f(x) = 0}, all of its limit points and is a closed subset of R. 0}. So there is a sequence {yn} such that {r : f(x) = 0} contains SO

Let f be a continuous function from R to R. Prove that { : f(x) = 0} is a closed subset of R. Solution. Let y be a limit point of {r : f (x) Yn E {x : f(x) = 0} for all n and limn-o Yn = y. Since f is continuous, by Theorem 40.2 we have f(y) = limn- f(yn) = lim,. 0 = 0. Hence y E {r : f(x) = 0}, all of its limit points and is a closed subset of R. 0}. So there is a sequence {yn} such that {r : f(x) = 0} contains SO

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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

What is a limit point? Please explain in detail.

38.6. Let f be a continuous function from R to R. Prove that {x : f(x) = 0} is a closed subset of R.
Solution. Let y be a limit point of {x : f(x)
Yn E {x : f(x) = 0} for all n and limn-o Yn = y. Since f is continuous, by Theorem 40.2 we
have f(y) = lim,→∞ f(Yn) = limn¬ 0 = 0. Hence y E {x : f(x) = 0}, so {x : f(x) = 0} contains
all of its limit points and is a closed subset of R.
0}. So there is a sequence {yn} such that

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