Variational principles: summary and problems

2.1 Differentiability and first order conditions If a function f : R → R has partial derivatives ∂if(x) = limt→0 t−1(f(x + tei) − f(x)) which exist and are continuous on R, it is a C1(R) function, and is differentiable at every x in the sense that f(x + h) − f(x) −∇f(x) · h = o(‖h‖) as h → 0. This means it can be approximated linearly, and the derivative is the linear map on R given by Df(x)(h) = ∇f(x) · h, which is linear in h.