2023-3-20

练习题 8.7

2 设

\[f(x,y)=\frac{1}{1-xy}~~~((x,y)\in[0,1]^2\setminus\{(1,1)\}). \]

求证: \(f\) 连续但不一致连续

\(f(x,y)\) 在 \([0,1]^2\setminus\{(1,1)\}\) 上显然连续.

任取 \(0<\varepsilon<1\) ,对于任意的 \(0<\delta<1\) ,令 \(x_1=1-\sqrt{\frac{\delta}{\varepsilon}},x_2=x_1+\delta\) ,

\(|f(x_1,1)-f(x_2,1)|=|\frac{1}{1-x_1}-\frac{1}{1-x_2}|=|\frac{x_1-x_2}{(1-x_1)(1-x_2)}|=\delta\cdot\frac{1}{(1-x_1)(1-x_2)}>\delta\cdot\frac{1}{(1-x_1)^2}=\varepsilon\) ,

故 \(f\) 不一致连续.

3 设 \(A\subset\R^n,\boldsymbol{p}\in\R^n\) .定义

\[\rho(\boldsymbol{p},A)=\inf\limits_{\boldsymbol{a}\in A}||\boldsymbol{p}-\boldsymbol{a}||, \]

称之为点 \(\boldsymbol{p}\) 到集合 \(A\) 的距离.证明:

(1) 若 \(A\neq\varnothing\) ,则 \(\overline{A}=\{\boldsymbol{p}\in\R^n:\rho(\boldsymbol{p},A)=0\}\) ;

(2) 对任何 \(\boldsymbol{p},\boldsymbol{q}\in\R^n\) ,有

\[|\rho(\boldsymbol{p},A)-\rho(\boldsymbol{q},A)|\leq||\boldsymbol{p}-\boldsymbol{q}||. \]

​ 这说明 \(\rho(\boldsymbol{p},A)\) 是 \(\R^n\) 上的连续函数.

(1) \(\rho(\boldsymbol{p},A)=0\Leftrightarrow\forall r>0,\exist\boldsymbol{a}\in B_r(\boldsymbol{p})\Leftrightarrow \boldsymbol{p}\in\overline{A}\) ,所以自然有 \(\overline{A}=\{\boldsymbol{p}\in\R^n:\rho(\boldsymbol{p},A)=0\}\).

(2) \(\exists\boldsymbol{a}\in\overline{A}\) ,使 \(\rho(\boldsymbol{p},A)=||\boldsymbol{p}-\boldsymbol{a}||\) .所以 \(|\rho(\boldsymbol{p},A)-\rho(\boldsymbol{q},A)|=|\,||\boldsymbol{p}-\boldsymbol{a}||-\rho(\boldsymbol{q},A)|\leq|\,||\boldsymbol{p}-\boldsymbol{a}||-||\boldsymbol{q}-\boldsymbol{a}||\,|\leq||\boldsymbol{p}-\boldsymbol{q}||\).

​ 故 \(\rho(\boldsymbol{p},A)\) 在 \(\R^n\) 上一致连续.

4 设 \(A,B\subset\R^n\) .定义

\[\rho(A,B)=\inf\{||\boldsymbol{p}-\boldsymbol{q}||:\boldsymbol{p}\in A,\boldsymbol{q}\in B\}, \]

称之为集合 \(A\) 和 \(B\) 之间的距离.证明:

(1) 若 \(A\) 为紧致集,则存在一点 \(\boldsymbol{a}\in A\) ,使得 \(\rho(\boldsymbol{a},B)=\rho(A,B)\) ;

(2) 若 \(A,B\) 为紧致集,则存在点 \(\boldsymbol{a}\in A,\boldsymbol{b}\in B\) ,使得 \(||\boldsymbol{a}-\boldsymbol{b}||=\rho(A,B)\) ;

(3) 设 \(A\) 为紧致集, \(B\) 为闭集,则 \(\rho(A,B)=0\) 当且仅当 \(A\cap B\neq\varnothing\) .

(1) 显然 \(\rho(A,B)=\inf\{\rho(\boldsymbol{a},B):\boldsymbol{a}\in A\}\).故若 \(\{\rho(\boldsymbol{a},B):\boldsymbol{a}\in A\}\) 为紧致集,则必存在 \(\rho(\boldsymbol{a},B)=\rho(A,B)\) ;

​ 而 \(A\) 为紧致集, \(\rho(\boldsymbol{a},B)\) 在 \(\R^n\) 上连续,故 \(\{\rho(\boldsymbol{a},B):\boldsymbol{a}\in A\}\) 紧致.证毕.

(2) \(B\) 为紧致集, \(f(\boldsymbol{b})=||\boldsymbol{b}-\boldsymbol{a}||\) 在 \(\R^n\) 上连续,所以 \(\{f(\boldsymbol{b}):\boldsymbol{b}\in B\}\) 紧致,

​ 于是有 \(||\boldsymbol{a}-\boldsymbol{b}||=\rho(\boldsymbol{a},B)=\rho(A,B)\).

(3) \(\exists\boldsymbol{a}\in A\) ,使 \(\rho(A,B)=\rho(\boldsymbol{a},B)=0\) ,故 \(\boldsymbol{a}\in\overline{B}\) ,而 \(B\) 为闭集,所以 \(\boldsymbol{a}\in B\) , \(\boldsymbol{a}\in A\cap B\neq\varnothing\).