Solving of partial differential equations under minimal conditions

It is proved that a differentiable with respect to each variable function f : R-2 -> R is a solution of the equation partial derivative u/partial derivative x + partial derivative u/partial derivative y = 0 if and only if there exists a function phi : R -> R such that f(x,y) = phi(x-y). This gives a positive answer to a queation by R. Baire. Besides, this result is used to solve analogous partial differential equations in abstract spaces and partial differential equations of higher-order.