  Let $f:X\to X$ be a continuous map on a compact metric space with finite
topological entropy. Further, we assume that the entropy map $\mu\mapsto
h_\mu(f)$ is upper semi-continuous. It is well-known that this implies the
continuity of the localized entropy function of a given continuous potential
$\phi:X\to R$. In this note we show that this result does not carry over to the
case of higher-dimensional potentials $\Phi:X\to R^m$. Namely, we construct for
a shift map $f$ a $2$-dimensional Lipschitz continuous potential $\Phi$ with a
discontinuous localized entropy function.
