  Let $S$ be a noetherian scheme and $f\colon X\to S$ be a smooth morphism of
relative dimension 1. For a locally constant sheaf on the complement of a
divisor in $X$ at over $S$, Deligne and Laumon proved that the universal local
acyclicity is equivalent to the local constancy of Swan conductors. In this
article, assuming the universal local acyclicity, we show an analogous result
of the continuity of local epsilon factors. We also give a generalization of
this result to a family of isolated singularities.
