  Let $(\mu_{\alpha})$ be a net of Radon sub-probability measures on the real
line, and $(t_{\alpha})$ be a net in $]0,+\infty[$ converging to 0. Assuming
that the generalized log-moment generating function $L(\lambda)$ exists for all
$\lambda$ in a nonempty open interval $G$, we give conditions on the left or
right derivatives of $L_{\mid G}$, implying vague (and thus narrow when $0\in
G$) large deviations. The rate function (which can be nonconvex) is obtained as
an abstract Legendre-Fenchel transform. This allows us to strengthen the
G\"{a}rtner-Ellis theorem by removing the usual differentiability assumption. A
related question of R. S. Ellis is solved.
