Let f be a locally Lipschitz function on a Banach space X, and S a subset of X. We define regular (i.e. noncritical) points for f relative to S, and give a sufficient condition for a point
z
∈
S
z \, \in \, S
to be regular. This condition is then expressed in the particular case when f is
C
1
{C^1}
, and is used to obtain a new proof of Hoffman’s inequality in linear programming.