It is well-known that any bijection $\mathbb{R} \rightarrow \mathbb{R}^2$ cannot be continuous. But suppose we have the two spaces $A = \{f(x):\mathbb{R^2}\rightarrow \mathbb{R} \}$ and $B = \{f(x):\mathbb{R}\rightarrow \mathbb{R} \}$ and and a map between them $A \xrightarrow{\phi} B$. (perhaps adding regularity conditions on the functions in $A,B$ so they're Hilbert or Banach spaces and continuity can be defined for $\phi$). The question is, can $\phi$ be both continuous and bijective?
This question is inspired by trying to invert the Radon transform for tensor fields, as in https://arxiv.org/pdf/1311.6167.pdf, but can be formulated outside of this context.
EDIT: The conditions I want on $A,B$ should be function spaces of continuously differentiable functions, or even stronger like analytic functions, equipped with some $L^p$ norm. These regularity conditions are to model something like the Radon or tensor transform in the sense ensuring that small perturbations to the data lead to small perturbations in the reconstructions.