  We introduce first-time sensitivity for a homeomorphism of a compact metric
space, that is a condition on the first increasing times of open balls of the
space. Continuum-wise expansive homeomorphisms, the shift map on the Hilbert
cube, and also some partially hyperbolic diffeomorphisms satisfy this
condition. We prove the existence of local unstable continua satisfying similar
properties with the local unstable continua of cw-expansive homeomorphisms, but
assuming first-time sensitivity. As a consequence we prove that first-time
sensitivity (with some additional technical assumptions) implies positive
topological entropy.
