  The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz
multiplicities in families of ideals. It is shown that Hilbert-Samuel
multiplicity is upper semicontinuous almost generally and that Hilbert-Kunz
multiplicity is upper semicontinuous in families of finite type. Surprisingly,
our machinery can be applied for families over Z and yields a partial solution
to the question about characteristic zero Hilbert-Kunz multiplicity posed by
Brenner, Li, and Miller. Another application is that for an affine ring the
infimum in the definition of F-rational signature, an invariant defined by
Hochster and Yao, is attained.
