  We investigate the properties of minimizers of one-dimensional variational
problems when the Lagrangian has no higher smoothness than continuity. An
elementary approximation result is proved, but it is shown that this cannot be
in general of the form of a standard Lipschitz "variation". Part of this
investigation, but of interest in its own right, is an example of a nowhere
locally Lipschitz minimizer which serves as a counter-example to any putative
Tonelli partial regularity statement. Under these low assumptions we find it
nonetheless remains possible to derive necessary conditions for minimizers, in
terms of approximate continuity and equality of the one-sided derivatives.
