  Let $f:C\rightarrow D$ be a nonconstant separable morphism between
irreducible smooth projective curves defined over an algebraically closed
field. We say that $f$ is genuinely ramified if ${\mathcal O}_D$ is the maximal
semistable subbundle of $f_*{\mathcal O}_C$ (equivalently, the homomorphism of
etale fundamental groups is surjective). We prove that the pullback
$f^*E\rightarrow C$ is stable for every stable vector bundle $E$ on $D$ if and
only if $f$ is genuinely ramified.
