A high-pass or low-pass filter's purpose is to reduce or eliminate spectral information content in portions of the stop-band. Therefore, depending on the frequency (near or at a zero), some spectral information will no longer exist in the result (or perhaps be reduced to numerical noise).
An inverse filter that attempts to correctly re-create some original ...
the inverse filter has a pole at z=-1.
A high pass filter has a zero at $z=1$ hence the inverse should have the pole at $z=1$. Both high pass and inverse high pass should have unity gain at $z=-1$,
But yes, the inverse is unstable.
in that case, how can i design such a digital filter so that it is stable and gives me the same response as expected,