The discussion establishes that if the partial derivatives of a function \( f: \mathbb{R}^n \to \mathbb{R}^n \) are bounded on a ball \( B_r(p) \), then \( f \) is Lipschitz continuous on that ball. Utilizing the Mean Value Theorem for partial derivatives, it is shown that there exists a point \( c \) in \( B_r(p) \) such that \( f'(c)(x) = \frac{f(x) - f(c)}{||x-c||} \) for all \( x \) in \( B_r(p) \). By defining \( M \) as the maximum of the bounded partial derivatives, the Lipschitz condition \( |f(x) - f(c)| \leq M||x-c|| \) is satisfied, confirming that \( f \) has a Lipschitz constant \( M \).
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