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Math Help - Continuous Function Question 2

  1. #1
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    Continuous Function Question 2

    Suppose functions f,g : (a,b) -> R are continuous, and f(r) = g(r) for each rational number r in (a,b). Show that f(x) = g(x) for each x element (a,b).

    Assume f(x) not equal g(x) for some x0 element (a,b) and a<x0<b
    Since both functions are continuous we have:
    lim(xf) = x0 and limf(xf) = f(x0)
    lim(xg) = x0 and limg(xg) = g(x0)
    This must hold true for x0 = r since a<r<b. Then, f(r) = g(r).

    I think i'm making some leap somewhere can anyone help me clarify what i'm trying to do if by chance it even remotely looks like anything good.
    Thanks
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  2. #2
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    Suppose that f(x_0 ) \ne g(x_0 ) . Every real number is the limit of a sequence of rational numbers. So there is a sequence of rational numbers in (a,b) and \left( {r_n } \right) \to x_0 . But is it possible for \left( {f\left( {r_n } \right)} \right) \to f\left( {x_0 } \right) and \left( {g\left( {r_n } \right)} \right) \to g\left( {x_0 } \right) given that f(r_n ) = g(r_n )
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