# Why aren't the integrator and the differentiator inverse systems?

I have a statement that leads to a paradox, but I'm incapable of finding the part where I'm wrong.

The integrator system

$$x(t) \mapsto y(t)=\int_{-\infty}^{t}{x(\tau) \, {\rm d} \tau}$$

is a linear time-invariant (LTI) system. Also, the differentiator

$$x(t) \mapsto y(t)=\frac{dx(t)}{dt}$$

is a linear time-invariant (LTI) system due to the fact that the system acts as the $$D$$ operator and the $$D$$ operator is LTI.

The inverse system of the integrator is the differentiator, and due to the commutative property of convolution, the integrator must be the inverse function of the differentiator but this is wrong since giving inputs $$x_1(t)=t$$ and $$x_2(t)=t+5$$ to the cascade of differentiator and integrator produces the same output. In other words, the integrator is incapable of retrieving the constant term of the input.

Well, what is wrong in this reasoning?

• Exhaustive answer over on math.SE.
– Peter K.
Jun 29, 2021 at 19:13
• @PeterK. What I'm looking for is more of a system view of the problem, namely the part that the flaw occurs, this should be in either linearity and time-invariance of differentiator and integrator or in properties of convolution. Now I know that some informality such as generalized functions is introduced in signals and systems, but unless we find the part where this informality is causing this paradox, we can't be sure of the other conclusions we draw using these properties. Jun 29, 2021 at 19:31