Section 18: Problem 12 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 $F:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be defined by the equation $F(x\times y)=\begin{cases} xy/(x^{2}+y^{2}) & \mbox{ if }x\times y\neq0\times0.\\ 0 & \mbox{ if }x\times y=0\times0. \end{cases}$

(a) Show that $F$ is continuous in each variable separately.

(b) Compute the function $g:\mathbb{R}\rightarrow\mathbb{R}$ defined by $g(x)=F(x\times x)$ .

(a) If $y_{0}=0$ then $F(x,y_{0})=0$ is continuous, otherwise $F(x,y_{0})=xy_{0}/(x^{2}+y_{0}^{2})$ is continuous as well.

(b) $g(x)=\begin{cases} \tfrac{1}{2} & \mbox{ if }x\neq0\\ 0 & \mbox{ if }x=0 \end{cases}$ .

(c) $F(x\times y)=\tfrac{1}{2}$ implies $x\times y\neq0\times0$ and $(x-y)^{2}=0$ . Therefore, $F^{-1}(\{\tfrac{1}{2}\})=\{(x,x)|x\neq0\}$ is not closed.

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Section 18: Problem 12 Solution

Section 18: Problem 12 Solution