Let f, g be continuous from R to R (the reals), and suppose that f(r) = g(r) for all rational numbers r. Is it true that f(x) = g(x) for all x [tex]\in[/tex] R?
The Attempt at a Solution
Basically, this seems trivial, but is probably tricky after all. I know that for f(x) to equal g(x) would mean that f(q) = g(q) where q is irrational as well as f(r) = g(r) as stated. I cannot think of example functions that are uniformly continuous on the Real line where this would fail, but yet, I also cannot think of a way to empirically prove that this is always true. Any help or a good starting point beyond this would be greatly appreciated. Note - this question follows the section of my text on "Combinations of Continuous Functions" but since this doesn't actually seem to combine f and g, beyond possibly the fact that f(x) = g(x) [tex]\Rightarrow[/tex] f(x) - g(x) = a continuous function h(x) as f, g continuous, I don't know of any other useful info in the text through this section.