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Let $X$ be a smooth projective curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p\>0$. Given a semistable vector bundle $E$ over $X$, we show that its direct image $F_{*}E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank-$p$ vector bundles $F_{*}L$, where $L$ is a line bundle over $X$.