Welche dieser Funktionen f: ℝ2 → ℝ ist stetig? Und warum? Wie zeig ich das?
(a)
\( f(x, y)=\left\{\begin{array}{ll} \frac{y^{2}}{x} & x \neq 0 \\ 0 & x=0 \end{array}\right. \)
(b)
\( f(x, y)=\left\{\begin{array}{ll} \frac{\phi(x)-\phi(y)}{x-y} & x \neq y \\ \phi^{\prime}(x) & x=y \end{array}\right. \)
c)
\( f(x, y)=\left\{\begin{array}{ll} \frac{x y}{x^{2}+y^{2}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0) \end{array}\right. \)
d)
\( f(x, y)=\left\{\begin{array}{ll} \frac{x_{4}}{x^{4}+y^{4}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0) \end{array}\right. \)
e)
\( f(x, y)=\left\{\begin{array}{ll} \frac{x^{2} y}{x^{4}+y^{2}} & (x, y) \neq(0,0) \\ 0 & (x, y)=(0,0) \end{array}\right. \)