  It was established in [8] that Lipschitz inf-compact functions are uniquely
determined by their local slope and critical values. Compactness played a
paramount role in this result, ensuring in particular the existence of critical
points. We hereby emancipate from this restriction and establish a
determination result for merely bounded from below functions, by adding an
assumption controlling the asymptotic behavior. This assumption is trivially
fulfilled if $f$ is inf-compact. In addition, our result is not only valid for
the (De Giorgi) local slope, but also for the main paradigms of average descent
operators as well as for the global slope, case in which the asymptotic
assumption becomes superfluous. Therefore, the present work extends
simultaneously the metric determination results of [8] and [18].
