real analysis help: continuous functions pf?

Let f be a continuous function defined on (a, b). Supposed f(x)=0 for all rational numbers x in (a, b). Prove that f(x)=0 on (a, b).

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  • 8 years ago
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    Let c be any irrational number such that a < c < b. Since the rationals are dense in IR, we can find a sequence {c_n} of rational numbers in (a, b) such that c_n -> c.

    As f is continuous, f preserves convergent sequences. Hence

    lim f(c_n) = f(c).

    n->∞

    As f(c_n) = 0 for all n, we have f(c) = 0. As c was arbitrary, we can conclude that f(x) = 0 for all irrational numbers in (a, b). Hence f(x) = 0 for all x in (a, b).

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