Here is the Fourier inversion theorem page in Wikipedia.
It states that for every function $f(x)$ that satisfy some conditions ($f(x)$ can be a function such as a Schwartz function, an integrable function in one dimension, or a square integrable function), $\mathcal F^{-1} (\mathcal F f) = f$ where $\mathcal F$ is the Fourier transform.
What I want to find is a function $g(x)$ such that $\mathcal F^{-1} (\mathcal F g) \neq g$. The conditions in the Wikipedia page seem to be somewhat sufficient conditions, and I can't come up with such $g(x)$.
Is there any conceivable function $g(x)$?