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is the following system invertible?

the system y

as I understand it, invertible means finding an inverse function which should return back the original input from an output of the given system.

if so I understand correct. Then one could say, yes the system is invertible and the inverse function is y(t) inverse

but for example, If our original input is x + 3 then its derivative is 1,
and if I try to use the inverse function it returns x + c.

is the nature of not knowing what exactly c equals makes the system non-invertible; meaning the original input cannot be restored. Does that make the system non-invertible

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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.

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