  Let $X$ denote an integral, projective Gorenstein curve over an algebraically
closed field $k$. In the case when $k$ is of characteristic zero, C. Widland
and the second author have defined Weierstrass points of a line bundle on $X$.
In the first section, this definition is extended to linear systems in
arbitrary characteristic. This definition may be viewed as a generalization of
the definitions of Laksov and St\"ohr-Voloch to the Gorenstein case. In the
second section, an example is given to illustrate the definition. This example
is a plane curve of arithmetic genus 3 in characteristic 2 such that the gap
sequence at every smooth point (with respect to the dualizing bundle) is 1,2,5
and there are no smooth Weierstrass points. In the third section, the
Weierstrass weight of a unibranch singularity (on a not necessarily rational
curve) is computed in terms of its semigroup of values. In the final section,
the Weierstrass weight of a singularity with precisely two branches (again
assuming that the characteristic is zero) is computed.
  AMS Classification: Primary 14H55, Secondary 14H20.
