Use of quotient space to turn differentiation into a bijective operator

Question

Not sure if this has been asked before. Recently, I've been thinking about differentiation and integration as linear operators acting on functional spaces. I know that integration cannot be properly described as the inverse of differentiation , as the latter is not injective. Still, it came to me that it may be possible to treat differentiation as a bijective operator, with integration as its inverse, if the domain is restricted to a particular quotient space that I will now outline. Consider the application $$\mathcal D:C^1(I) \to C^0(I), \qquad \mathcal D(f(x))=f'(x) \quad \forall x \in I,$$ where I have chosen the domain to be the space of differentiable real-valued functions with continuous derivative on the interval $I \subseteq \Bbb R$ for simplicity. Now to my question. Consider the equivalence relation $$r = \{(f,g) \in C^1(I) \times C^1(I) \ | \ \forall x \in I \quad f(x) = g(x)+k, \ k \in \Bbb R\}$$ and the resulting quotient space $C^1(I)/r$. In this new set, each function belongs to a unique equivalence class in which all elements differ by a constant. If we then restrict the domain of $\mathcal D$ to said quotient space, would the application become bijective, with integration (in the indefinite sense) as its inverse? And could this work for broader functional spaces as well?