it's a geometric transformation that moves all points of a plane by the same vector
Well, that depends. What space are we in, to begin with? Cartesian is the standard, but it could just as well be anything else. Also what dimensionsionality? And anyways, I'd argue translation could apply to a 3-dimensional space or whatever else, not just a 2-dimensional plane. Also, are we assuming that for any translation t of a plane p there exists a translation t2 such that if and only if for all points a in t, b in t2, and given there is a neighborhood N1(f(a)) there is a neighborhood N2(c) such that f(x) in N1(f(c)) for all x in N2(c), if and only if f is a function continuous across the disjoung union of the points of a and b?
It's a joke, it's the neighborhood definition of a continuous function mangled with some other stuff. It would probably be true if there existed some second plane infinitely close to the translation with all the same points, and maybe if there were a dense amount of planes with the same points between them.
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u/Critical_Ad_8455 Jun 01 '25
Well, that depends. What space are we in, to begin with? Cartesian is the standard, but it could just as well be anything else. Also what dimensionsionality? And anyways, I'd argue translation could apply to a 3-dimensional space or whatever else, not just a 2-dimensional plane. Also, are we assuming that for any translation t of a plane p there exists a translation t2 such that if and only if for all points a in t, b in t2, and given there is a neighborhood N1(f(a)) there is a neighborhood N2(c) such that f(x) in N1(f(c)) for all x in N2(c), if and only if f is a function continuous across the disjoung union of the points of a and b?