Section 18: Problem 6 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

Find a function $f:\mathbb{R}\rightarrow\mathbb{R}$ that is continuous at precisely one point.

For example, $f(x)=x$ if $x\in\mathbb{Q}$ , or $0$ otherwise. For every $x$ , the image of the open interval $I_{\epsilon}=(x-\epsilon,x+\epsilon)$ , $\epsilon>0$ , is $(I_{\epsilon}\cap\mathbb{Q})\cup\{0\}$ . So, if $x\neq0$ , then, regardless of the value of $f(x)$ , for no $\epsilon>0$ the image of $I_{\epsilon}$ is a subset of, say, $(f(x)-|x|,f(x)+|x|)$ . But if $x=0$ , then for every $\epsilon<\delta$ , the image of $I_{\epsilon}$ is a subset of $I_{\delta}$ .

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Section 18: Problem 6 Solution

Section 18: Problem 6 Solution