We study the closure properties of the function classes GapP and GapP_+. We characterize the property of GapP_+ being closed under decrement and of GapP being closed under maximum, minimum, median, or division by seemingly implausible collapses among complexity classes; thereby giving evidence that these function classes don't have these closure properties.

We show a similar result concerning operations
we call bit cancellation and bit insertion:
Given a function f in GapP and
a polynomial-time computable function k.
Then we ask whether the function f^*(x)
that is obtained from f(x)
by canceling the k(x)th bit in the binary representation of f(x),
or whether the function f^+(x)
that is obtained from f(x)
by inserting a bit at position k(x) in the binary representation of f(x),
is also in GapP.
We give necessary conditions and a sufficient conditions
for GapP being closed under bit cancellation and bit insertion, respectively.