Suppose a function fa,b) -> R is continuous and f(r) = 0 for each rational number r in (a,b). Show that f(x) = 0 for each x element (a,b).
So by the definition of continuous every sequence (xn) element dom(f) converges to x0 and limf(xn) = f(x0). Then, xn = 0 must converge to x0. Then, limf(0) = lim0 = 0 = f(x0).
I was wondering if this was correct logic and if there other xn sequences that could be used.
Thanks