Name a function with a domain of all real numbers that is continuous only at a single point.
Definition: A function is continuous at point B if and only if the limit of f(x) as x approaches B is equal to f(B).
(In reply to
Non-piecewise by Jer)
I don't understand what you mean by "piecewise." The function Kereki
gives is F(x)=Id(x)*D(x) where Id(x)=x ("identity function")and D(x)=0
if x is rational, 1 if x is irrational ("Dirichlet function"). Usually
a "piecewise" function has different analytic definitions on different
intervals, but no intervals are involved in Kereki's function. So
Kereki's function fulfills your requirement of being non-piecewise, in
my book, anyway. A function not using the rationals can be designed
along similar lines -- the rationals can be replaced by other sets that
are dense in the real numbers (such sets do exist).
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Posted by Richard
on 2005-12-15 14:57:49 |
re: Non-piecewise
