You have a point, in that this is the argument of constructive mathematicians. If we remove the noncomputable points, then the intermediate value theorem is false. But a constructivist patches the problem by restricting continuous functions to computable functions and then the "constructive IVT" is true again. So you are in agreement with constructivism, the philosophy of math that says that every mathematical object must be computable (or constructible in some other sense. There are various flavors of constructivism as I understand it, but I'm not too familiar with the subject). https://en.wikipedia.org/wiki/Constructivism_(mathematics) A few mathematicians and physicists have attempted to frame modern physics in terms of constructive mathematics, but with very limited results to date. That's not to say we won't all feel differently about these matters in the future. I found an interesting looking paper on the subject, which I didn't read. https://arxiv.org/pdf/0805.2859.pdf. Pulling one bolded quote from the author's introduction: "algorithms must replace formulas." I trust this viewpoint would be satisfying to you. It's not mainstream but that doesn't make it wrong. The problem with mathematical constructivism is that it denies uncountable sets. So it throws out the last 140 years of mainstream mathematics; and with it, the conventional foundations of both classical and modern physics.