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