I'm sorry, but classical mathematics is perfectly able to handle continuous functions. And in fact it does so more easily. Which is why most of the important theorems were proven classically first.
Sure, Errett Bishop came along with https://www.amazon.com/Foundations-Constructive-Analysis-Err... and fixed that. But it is harder. And sure, as soon as you start looking at error bounds in numerical analysis, the classical shortcuts start to take work. But it is simply wrong to assert that the continuous REQUIRES the law of the excluded middle.
In fact every mathematician has been through the classical treatments of continuity in courses on real analysis and topology. Very few can say much that is sensible about constructivism.
Sure, Errett Bishop came along with https://www.amazon.com/Foundations-Constructive-Analysis-Err... and fixed that. But it is harder. And sure, as soon as you start looking at error bounds in numerical analysis, the classical shortcuts start to take work. But it is simply wrong to assert that the continuous REQUIRES the law of the excluded middle.
In fact every mathematician has been through the classical treatments of continuity in courses on real analysis and topology. Very few can say much that is sensible about constructivism.