Section 18: Problem 1 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

Prove that for functions $f:\mathbb{R}\rightarrow\mathbb{R}$ , the $\epsilon-\delta$ definition of continuity implies the open set definition.

Let $V\subset\mathbb{R}$ be open and $f$ be continuous according to the $\epsilon-\delta$ definition. For every $x\in f^{-1}(V)$ , $f(x)=y\in V$ and $\exists$ $\delta>0$ such that $(y-\delta,y+\delta)\subset V$ . Further, there is $\epsilon>0$ such that $f((x-\epsilon,x+\epsilon))\subset(y-\delta,y+\delta)$ implying $(x-\epsilon,x+\epsilon)\subset f^{-1}(V)$ . Therefore, $f^{-1}(V)$ is open.

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Section 18: Problem 1 Solution

Section 18: Problem 1 Solution