Friedgut's theorem for the continuous cube

A celebrated theorem of
Friedgut says that every boolean function on the discrete cube can be
approximated by a function which depends only on a number of variables
that depends on the sum of the influences of the variables of f. Dinur
and Friedgut conjectured an analogue of this theorem for the continuous
cube. I disprove their conjecture, and then prove the correct version of
Friedgut's theorem for the continuous cube.