If your Math professor claims you failed a midterm, then you show them they forgot to count a page's worth of marks, but they said you still the same grade because they doubted your "willingness" to pass, you'd be well within your rights to complain.
I really don't think that it is as simple as that. If your professor refused to write you a recommendation for a research position because your grades are a bit low and he doesn't think you are responsible because you were dangerously close to missing important deadlines, then you show that his grade calculation was wrong but he still won't recommend you because the second part was still true.
