In this paper classical problem of the calculus of variations is investigated assuming that the integrand is a continuous function. The authors apply the main ideas of non-smooth analysis in order to prove a non-smooth version of the classical Euler equation. The usual assumption of existence of a common \( L^1 \)-upper bound of a family of summable functions is replaced by uniform integrability of the same family. The proposed in the paper technique do not use variational principles. A necessary optimality condition for the basic problem of calculus of variations is obtained.