If $f: \mathbb{R}\setminus\mathbb{Q} \to \mathbb{R}_S\times\mathbb{R}_S$ is continuous, then the image has empty interior.

HINT: Your idea of using the (reverse) diagonal is a good one. Any line segment $L$ in the plane with slope $-1$ is an uncountable closed, discrete set in the Sorgenfrey plane. What does that tell you about $f^{-1}[L]$ in the irrationals?