Occurrence of gap for one-dimensional scalar autonomous functionals with one end point condition

Let $L:\mathbb{R}\times \mathbb{R}\to [0,+\mathrm{\infty }[\cup \{+\mathrm{\infty }\}$ be a Borel function. We consider the problem \begin{equation}\tag{P}\min F(y)=\int_0^1L(y(t), y’(t)),dt: y(0)=0,, y\in W^{1,1}([0,1],\mathbb R).\end{equation} We give an example of a real valued Lagrangian $L$ for which the Lavrentiev phenomenon occurs. We state a condition, involving only the behavior of $L$ on the graph of two functions, that ensures the non-occurrence of the phenomenon. Our criterium weakens substantially the well-known condition, that $L$ is bounded on bounded sets.