  If $X$ is an almost complex manifold, with an almost complex structure $J$ of
class $\CC^\alpha$, for some $\alpha >0$, for every point $p\in X$ and every
tangent vector $V$ at $p$, there exists a germ of $J$-holomorphic disc through
$p$ with this prescribed tangent vector. This existence result goes back to
Nijenhuis-Woolf. All the $J$ holomorphic curves are of class $\CC^{1,\alpha}$
in this case.
  Then, exactly as for complex manifolds one can define the Royden-Kobayashi
pseudo-norm of tangent vectors. The question arises whether this pseudo-norm is
an upper semi-continuous function on the tangent bundle. For complex manifolds
it is the crucial point in Royden's proof of the equivalence of the two
standard definitions of the Kobayashi pseudo-metric. The upper semi-continuity
of the Royden-Kobayashi pseudo-norm has been established by Kruglikov for
structures that are smooth enough. In [I-R], it is shown that $\CC^{1,\alpha}$
regularity of $J$ is enough.
  Here we show the following:
  Theorem. There exists an almost complex structure $J$ of class $\CC^{1\over
2}$ on the unit bidisc $\D^2\subset \C^2$, such that the Royden-Kobayashi
seudo-norm is not an upper semi-continuous function on the tangent bundle.
