# Invertibility of an ideal differentiator

Is the system $$y(t)= dx(t)/dt$$ invertible or not? If yes, please determine the inverse system for it.

It is not. It suffices to find a counter example. Let me tell you that "I am flat". Could you derive my actual value?

So, any constant function $$f(t) =c$$ is differentiable, and have the same derivative, $$f'(t)=0$$. Only from knowing that $$f'(t)=0$$, you cannot recover the original (constant) function. Hence the system is not invertible.

The system is not invertible because you can always add an arbitrary constant $$c$$ to any function $$x(t)$$ and the system will map it to same differentiated function $$y(t)$$. So, the mapping is not unique or one-to-one and hence not invertible.

Delta coding (with no quantization) is quite close to what you are asking. Let:

x=randi([0 10], 8,1); delta = diff([0; x]); x_hat = cumsum(delta); assert(all(x-x_hat)==0);

But then you have a «silent agreement» to preprepend a zero such that you have a known starting point. And differentiation/integration is replaced by its discrete equivalents.

Knut