Discussion
Let D be the rectangle :
\[ \{(x, y) \in \mathbf{R}^{2} \mid |x| \le 1, |y| \le 1\} \]
and \(h\) and \(g\) are functions defined on D given by :
\(h(x, y) = xy^{2}\) and
\(g(x, y) = y^{2/3}\).
Then :
Answer: A
Only \(h\) satisfies the Lipschitz condition on D.
The Lipschitz condition here is in \(y\): \(|F(x, y_{1}) - F(x, y_{2})| \le K|y_{1} - y_{2}|\). For \(h = xy^{2}\), \(\partial h/\partial y = 2xy\) and \(|2xy| \le 2\) on D, so by the mean value theorem \(K = 2\) works.
For \(g = y^{2/3}\), take \(y_{2} = 0\):
\[ \frac{|g(y) - g(0)|}{|y - 0|} = |y|^{-1/3} \to \infty \]
as \(y \to 0\), so no constant K works. This is the classic example where \(y' = y^{2/3}\) loses uniqueness. That rules out options naming \(g\) or neither.
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