Is Differentiation as a system, is an invertible system?
Yes, your understanding is correct. The differentiation is not an invertible system. And, the reason is precisely, because of the possible unknown constant which gets nulled by differentiation.

Basically, the differentiation of $f(t)$ and $(f(t) + c)$ is indistinguishable $f^{'}(t)$, which makes the system, many-to-one, and hence non-invertible.