- Continuity of transformations
- Journal of Mathematical Analysis and Applications
- Pages (from-to)
- Document type
- Faculty of Economics and Business (FEB)
- Amsterdam School of Economics Research Institute (ASE-RI)
Let u be a continuous function from Re to Re. Let f be a transformation (which in our terminology does not have to be bijective) from the range of u to Re. v = f(u(.)) is the composition of f and u. It is well-known that continuity of f implies continuity of v. We consider the reversed question: Does continuity of v imply continuity of f? Elementary as this question may be, we did not find a place in literature where the answer is given. In fact it is our experience that the probability that a mathematician at first sight will gamble on the wrong answer, is an increasing function of his familiarity with elementary analysis, and is always above 1/2. This paper will answer the reversed question above, in a somewhat more general setting, and give applications.
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