MHB Lip Functions: Partial Derivatives on Ball Br(p)

[onie mti]

Can i get an idea of how to show that if the partial derivates of the components of a R_n-R_n function f are boounded on a ball B_r(p) then f is Lip on the ballI defined f to be a R_n-R_n function defined on a set D

 

[Jhoneine]

containing Br(p), we must show that f is Lipschitz on Br(p). We can start by using the Mean Value Theorem for partial derivatives. This states that for each component of the function, there exists a point c in Br(p) such thatf'(c)(x) = (f(x) - f(c))/||x-c|| for all x in Br(p). By assumption, the partial derivatives of f are bounded on Br(p). Let M be the maximum of all the partial derivatives. Then, for all x in Br(p), |f(x)-f(c)| ≤ M||x-c||.This implies that f is Lipschitz on Br(p) with Lipschitz constant M.