We show that the conclusion of the Plachky-Steinebach theorem holds true for intervals of the form ](L) over bar (r)'(lambda), y[, where (L) over bar (r)'(lambda) is the right derivative (but not necessarily a derivative) of the generalized log-moment generating function (L) over bar with some lambda > 0 and y is an element of](L) over bar (r)'(lambda), +infinity], under only the following two conditions: (a) (L) over bar (r)'(lambda is a limit point of the set {(L) over bar (r)'(t): t > lambda}, and (b) (L) over bar (t(i)) has limit with t(i) belonging to some decreasing sequence converging to sup{ t > lambda: (L) over bar (]lambda,t]) is affine}. By replacing (L) over bar (r)'(lambda), +infinity], under only the following two conditions: (a) (L) over bar (r)'(lambda is a limit point of the set {(L) over bar (r)'(t): t > lambda}, and (b) (L) over bar (t(i)) has limit with t(i) belonging to some decreasing sequence converging to sup{t > lambda: (L) over bar (vertical bar]lambda,t]) is affine}. By replacing (L) over bar (r)'(lambda) with (L) over bar (r)'(lambda(+)), the above result extends verbatim to the case lambda = 0 (replacing (a) by the right continuity of (L) over bar at zero when (L) over bar (r)'(0(+)) = -infinity). No hypothesis is made on (L) over bar (]-infinity,lambda[) for example, ](L) over bar (]-infinity,lambda[ )may be the constant. +infinity when lambda = 0); lambda >= 0 may be a nondifferentiability point of (L) over bar and, moreover, a limit. point of nondifferentiability points of (L) over bar; lambda = 0 may be a left and right discontinuity point of (L) over bar. The map (L) over bar (vertical bar]lambda,lambda+epsilon[) may fail to be strictly convex for all epsilon > 0. If we drop the assumption (b), then the same conclusion holds with upper limits in place of limits. Furthermore, the foregoing is valid for general nets (mu(alpha),c(alpha)) of Borel probability measures and powers (in place of the sequence (mu(n), n(-1))) and replacing the intervals ](L) over bar (r)'(lambda(+)), y[ by ]x(alpha), y(alpha)],( )where (x(alpha), y(alpha)) is any net such that (x(alpha)) converges to (L) over bar (r)'(lambda(+)) and lim inf(alpha) y(alpha) > (L) over bar (r)'(lambda(+)).