$\mathbb{RP}^2$ does not embed into $\mathbb{R}^3$: reduction to the differentiable case

Question

It is not difficult to see that the real projective plane cannot be embedded into $\mathbb{R}^3$ as a differentiable submanifold (for example one can easily show that the complement would consist of two connected components, so the projective plane would be orientable, which is not the case).
However, it is also well-known that there is no continuous embedding. My question:
Is there a way to reduce the continuous case to the differentiable one, thus avoiding the use of homology or other advanced tools?