[math-fun] discontinuous rational functions & school textbooks
It actually is possible for a rational function OF SEVERAL VARIABLES to be discontinuous. I.e. different limits as you approach a 0/0 point from different directions. Little known fact. You might want to read Feynman about his experiences re school textbooks; he was on a California board trying to decide on textbooks. He seemed to conclude it was beyond his power to correct the idiocy.
Such examples, like f(x,y) = (x^2 - y^2)/(x^2 + y^2) (= cos(2θ)), are in a lot of textbooks, like Little Rudin (as of 1965), at least in the exercises. But I don't think most people would consider the point approached by the domain points (like (0,0) in the above example) as being part of the domain. --Dan
It actually is possible for a rational function OF SEVERAL VARIABLES to be discontinuous. I.e. different limits as you approach a 0/0 point from different directions. Little known fact.
participants (2)
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Dan Asimov -
Warren D Smith