A function $f:\mathbb R^{1,1}\to \mathbb R^{1,1}$ is "pseudocontinuous" if and only if:

$$f(x,y)\equiv(g(x),h(y)),$$
or
$$f(x,y)\equiv(h(y),g(x)),$$
for any pair of continuous functions $f,g:\mathbb R\to\mathbb R$.

Why? I list the sequence of lemmas which one should prove:

**Lemma 1**: We can redefine the pseudometric to $(x,y)\mapsto |xy|$ instead of $(x,y)\mapsto |x^2 - y^2|$.

We now switch to this pseudometric.

**Lemma 2**: A "cross" (that is, the union of a line parallel to the $x$ axis with a line parallel to the $y$ axis) necessarily gets mapped to another cross by any pseudocontinuous function $f$.

**Lemma 3**: A horizontal line or vertical line gets mapped to a horizontal or vertical line by pseudocontinuous $f$.

**Lemma 4**: For fixed $x$, the function $y \mapsto f(x,y)$ is continuous. The same is true for $x \mapsto f(x,y)$ for fixed $y$.

**Lemma 5**: The function $f$ is continuous everywhere.

**Theorem**: The above statement.