Show that 
i) satisfies a Lipschitz condition on any rectangle  and ;

ii) satisfies a Lipschitz condition on any strip  and ;

iii) does not satisfy a Lipschitz condition on the entire plane.

i) To show that \( f(x,y) = xy \) satisfies a Lipschitz condition on the rectangle \( a \leq x \leq b \) and \( c \leq y \leq d \), we need to demonstrate that there exists a constant \( L \) such that for all \( (x_1, y_1) \) and \( (x_2, y_2) \) in this rectangle, the following inequality holds:

\[ |f(x_1, y_1) - f(x_2, y_2)| \leq L \cdot \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} \]

For \( f(x, y) = xy \), we have:

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