In this work we study the structure of approximate solutions of variational problems with continuous integrands f:[0,~)xR n xR n ->R 1 which belong to a complete metric space of functions M. We do not impose any convexity assumption and establish the existence of an everywhere dense G δ -set F M such that each integrand in F has the turnpike property. This result is an extention of the main result of Zaslavski (Nonlinear Anal. 42 (2000) 1465) which was obtained for integrands f M which were convex as a function of the last argument (derivative).