Prove continuity of the algebraic operations on \(\mathbb{R}\), as follows. Use the metric \(d(a, b) = \abs{a - b}\) on \(\mathbb{R}\) and the metric on \(\mathbb{R}^2\) given by the equation
\[ \rho((x, y), (x_0, y_0)) = \max\ts{\abs{x - x_0}, \abs{y - y_0}}. \]
- Show that addition is continuous. [Hint: Given \(\epsilon\), let \(\delta = \epsilon/2\) and note that
\[ d(x + y, x_0 + y_0) \leq \abs{x - x_0} + \abs{y - y_0}.] \]
- Show that multiplication is continuous. [Hint: Given \((x_0, y_0)\) and \(0 < \epsilon < 1\), let
\[ 3\delta = \epsilon/(\abs{x_0} + \abs{y_0} + 1) \]
and note that
\[ d(xy, x_0y_0) \leq \abs{x_0}\abs{y - y_0} + \abs{y_0}\abs{x - x_0} + \abs{x - x_0}\abs{y - y_0}.] \]
Show that the operation of taking reciprocals is a continuous map from \(\mathbb{R} - \ts{0}\) to \(\mathbb{R}\). [Hint: Show the inverse image of the interval \((a, b)\) is open. Consider five cases, according as \(a\) and \(b\) are positive, negative, or zero.]
Show that the subtraction and quotient operations are continuous.
Solution
Throughout, \(\rho((x,y),(x_0,y_0))<\delta\) means \(\abs{x-x_0}<\delta\) and \(\abs{y-y_0}<\delta\).
1. (a) Addition is continuous.
Given \(\varepsilon>0\), let \(\delta=\varepsilon/2\). If \(\rho((x,y),(x_0,y_0))<\delta\), then \(\abs{(x+y)-(x_0+y_0)}\le\abs{x-x_0}+\abs{y-y_0}<2\delta=\varepsilon\).
2. (b) Multiplication is continuous.
Given \((x_0,y_0)\) and \(0<\varepsilon<1\), let \(\delta=\varepsilon/\bigl(3(\abs{x_0}+\abs{y_0}+1)\bigr)\), so \(\delta<1\). Since \(xy-x_0y_0=x_0(y-y_0)+y_0(x-x_0)+(x-x_0)(y-y_0)\), if \(\rho((x,y),(x_0,y_0))<\delta\) then \[ \abs{xy-x_0y_0}<\abs{x_0}\delta+\abs{y_0}\delta+\delta^2<(\abs{x_0}+\abs{y_0}+1)\delta=\varepsilon/3<\varepsilon. \]
3. (c) The reciprocal \(r(x)=1/x\) is continuous on \(\RR-\{0\}\).
Let \(x_0\ne0\) and \(\varepsilon>0\), and put \(\delta=\min\{\abs{x_0}/2,\ \varepsilon\abs{x_0}^2/2\}\). If \(\abs{x-x_0}<\delta\), then \(\abs x>\abs{x_0}/2\), so \[ \Bigl\lvert\frac1x-\frac1{x_0}\Bigr\rvert=\frac{\abs{x-x_0}}{\abs x\abs{x_0}}<\frac{2\delta}{\abs{x_0}^2}\le\varepsilon. \]
4. (d) Subtraction and the quotient \((x,y)\mapsto x/y\) on \(\RR\times(\RR-\{0\})\) are continuous.
Negation \(y\mapsto-y\) is continuous, since \(\abs{(-y)-(-y_0)}=\abs{y-y_0}\). Subtraction is addition composed with \(\operatorname{id}\times(-)\), and the quotient is multiplication composed with \(\operatorname{id}\times r\); a product of continuous maps is continuous by E-G4SRA, and steps 1, 2, 3 give the remaining factors.
5. Q.E.D.
Steps 1 through 4 prove (a) through (d).