  In a first result we prove that every continuous local triple derivation on a
JB$^*$-triple is a triple derivation. We also give an automatic continuity
result, that is, we show that local triple derivations on a JB$^*$-triple are
continuous even if not assumed a priori to be so. In particular every local
triple derivation on a C$^*$-algebra is a triple derivation. We also explore
the connections between (bounded local) triple derivations and generalised
(Jordan) derivations on a C$^*$-algebra.
