Suppose f:R-->R satisfies f(x+y)=f(x)+f(y) for every x,y in R. If f is continuous at some point p, prove that there is some constant a in R s.t. f(x)=ax for all x. that is, an additive function that is continuous at even one point is linear-and hence continuous on all of R.
*Remark: In the original statement I had to fix a s.t. f(ax)=ax, but that is trivial because f(0)=0. For a nonzero a, the condition in the book could not have been made generally. Say for f(x) = 2x.
2007-08-02
06:37:15
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3 answers
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asked by
guyava99
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Mathematics