Suppose thatf: (a,b) →Ris continuous and thatf(r) = 0 for ever rational numberrє (a,b). Prove thatf(x) = 0 for allxє (a,b).

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- May 3rd 2009, 03:16 AMbearej50Continuous functionsSuppose that
*f*: (*a*,*b*) →**R**is continuous and that*f*(*r*) = 0 for ever rational number*r*є (*a*,*b*). Prove that*f*(*x*) = 0 for all*x*є (*a*,*b*).

- May 3rd 2009, 03:23 AMInfophile
Hello,

For all reals, it exists a sequence of rationals such as .

Since is continuous then

And so...

:) - May 3rd 2009, 08:02 AMHallsofIvy
Very nice- and not at all what I was thinking of. I was thinking of a proof by contradiction. Suppose there exist some a such that . Let . Then for all , there exist rational x in the interval . For that x, but , contradicting the fact that f is continuous.