  We give a K-theory proof of the invariance under cobordism of the family
index. We consider elliptic pseudodifferential families on a continuous fibre
bundle with smooth fibres over a compact base space B, and define a notion of
cobordant families using K^1-groups on fibrations with boundary. We show that
the index of two such families is the same using properties of the push-forward
map in K-theory to reduce it to families on B x R^n.
