I don’t understand how it is possible to have such an axiom. Why can you not construct the function $f : x \mapsto I(x > 2)$ where $I$ is 1 for true, 0 for false. And then prove that the preimage of $(-1,0.5)$ is $(-\infty,2]$ which is not open?
In your case it is not provable that (x > 2 or x =< 2), that would require exactly the law of the excluded middle.
This is explained in one of the later slides.
Does it break down somewhere in the definition of continuity?