Our formulas clearly hold also for non-square matrices. All partial derivatives exist and are continuous everywhere. The matrix of derivatives are determined easily by computing the partial derivatives. Proof We need to compute If u is the zero vector, the limit clearly exists and equals zero. We have show that all directional derivatives exist.
Theorem Let f be a real-valued function defined on an open set .
If the partial derivatives , then f is continuous in E . Taking norms and using the triangle inequality, we obtain which proves the claim. For convex sets E , our proof shows that f is Lipschitz with constant nM . Our estimates hold for all p -norms on . They can be slightly improved by using the 1-norm. 5.