第五章 多元函数微分学 · 第5.1题

5.1.16 问下列函数是否在全平面连续, 为什么?

(1) $f\left( {x,y}\right) = \left\{ \begin{matrix} \frac{{x}^{2} - {y}^{2}}{{x}^{2} + {y}^{2}}, & {x}^{2} + {y}^{2} \neq 0, \\ 0, & {x}^{2} + {y}^{2} = 0; \end{matrix}\right.$

(2) $f\left( {x,y}\right) = \left\{ \begin{array}{ll} \frac{\sin \left( {xy}\right) }{x}, & x \neq 0, \\ y, & x = 0; \end{array}\right.$

(3) $f\left( {x,y}\right) = \left\{ \begin{array}{ll} \frac{{x}^{2}}{{y}^{2}}{\mathrm{e}}^{-\frac{{x}^{4}}{{y}^{2}}}, & y \neq 0, \\ 0, & y = 0; \end{array}\right.$

(4) $f\left( {x,y}\right) = \left\{ \begin{array}{ll} {y}^{2}\ln \left( {{x}^{2} + {y}^{2}}\right) , & {x}^{2} + {y}^{2} \neq 0, \\ 0, & {x}^{2} + {y}^{2} =