Even for integer-valued continuous $f: X \to $ Z, there are generically lots of solutions which are not coboundaries (of either integer or real-valued ct functions). For example, suppose that $\mu$ is ergodic, and is supported on a uniquely ergodic minimal subset of the mSFT. Then the GPS (Giordano, Putnam, Skau) results apply, and the dimension group invariant gives plenty of examples with nontrivial infinitesimals. Every nontrivial infinitesimal gives rise to an integer-valued continuous function $f$ which vanishes not only under integration at $\mu$. On the other hand, if the minimal subset has more than one ergodic measure, then there must be non-coboundaries that integrate to zero (although there need not be any integer-valued examples of these).

I suppose there are some that must be Lipschitz continuous ...

Even for integer-valued continuous $f: X \to $ Z, there are generically lots of solutions which are not coboundaries (of either integer or real-valued ct functions). For example, suppose that $\mu$ is ergodic, and is supported on a uniquely ergodic minimal subset of the mSFT. Then the GPS (Giordano, Putnam, Skau) results apply, and the dimension group invariant gives plenty of examples with nontrivial infinitesimals. Every nontrivial infinitesimal gives rise to an integer-valued continuous function $f$ which vanishes under integration at $\mu$. On the other hand, if the minimal subset has more than one ergodic measure, then there must be non-coboundaries that integrate to zero (although there need not be any integer-valued examples of these).

I suppose there are some that must be Lipschitz continuous ...

Even for integer-valued continuous $f: X \to $ Z, there are generically lots of solutions which are not coboundaries (of either integer or real-valued ct functions). For example, suppose that $\mu$ is ergodic, and is supported on a minimal subset of the mSFT. Then the GPS (Giordano, Putnam, Skau) results apply, and the dimension group invariant gives plenty of examples with nontrivial infinitesimals. Every nontrivial infinitesimal gives rise to an integer-valued continuous function $f$ which vanishes not only under integration at $\mu$ but also at integration at every other invariant measure supported on the minimal subset, but is not a coboundary.

I suppose there are some that must be Lipschitz continuous ...

Even for integer-valued continuous $f: X \to $ Z, there are generically lots of solutions which are not coboundaries (of either integer or real-valued ct functions). For example, suppose that $\mu$ is ergodic, and is supported on a uniquely ergodic minimal subset of the mSFT. Then the GPS (Giordano, Putnam, Skau) results apply, and the dimension group invariant gives plenty of examples with nontrivial infinitesimals. Every nontrivial infinitesimal gives rise to an integer-valued continuous function $f$ which vanishes not only under integration at $\mu$. On the other hand, if the minimal subset has more than one ergodic measure, then there must be non-coboundaries that integrate to zero (although there need not be any integer-valued examples of these).

I suppose there are some that must be Lipschitz continuous ...