# Thread: Continuous Function Question 2

1. ## 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

2. 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 $f(r_n ) = g(r_n )$