Section 21: Problem 9 Solution

Working problems is a crucial part of learning mathematics. No one can learn topology merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.

James R. Munkres

Let $f_{n}:\mathbb{R}\rightarrow\mathbb{R}$ be the function $f_{n}(x)=\frac{1}{n^{3}[x-(1/n)]^{2}+1}.$ See Figure 21.1. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be the zero function.

(a) Show that $f_{n}(x)\rightarrow f(x)$ for each $x\in\mathbb{R}$ .

(b) Show that $f_{n}$ does not converge uniformly to $f$ . (This shows that the converse of Theorem 21.6 does not hold; the limit function $f$ may be continuous even though the convergence is not uniform.)

(a) $f_{n}$ is positive and unimodal as the denominator is a positive parabola, the mode is at $x=\tfrac{1}{n}$ , hence, for $x\le0$ , $f_{n}(x)\le f_{n}(0)\rightarrow0$ , and for $x>0$ , $n\ge\tfrac{2}{x}$ , $f_{n}(x)\le f_{n}(\tfrac{2}{n})=f_{n}(0)\rightarrow0$ as well.

(b) $|f_{n}(\tfrac{1}{n})-f(\tfrac{1}{n})|=1$ .

See an example of a similar function in the solution of Exercise 8.

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Section 21: Problem 9 Solution

Section 21: Problem 9 Solution