What is the key to proving continuity using rational numbers?

[steelphantom]

Two problems, actually, but they are very similar. Here goes:

Homework Statement

Let f be a continuous real-valued function with domain (a, b). Show that if f(r) = 0 for each rational number r in (a, b,), then f(x) = 0 for all x in (a, b).

Homework Equations

The Attempt at a Solution

Homework Statement

Let f and g be continuous real-valued functions on (a, b) such that f(r) = g(r) for each rational number r in (a, b). Prove that f(x) = g(x) for all x in (a, b).

Homework Equations

The Attempt at a Solution

Alright, well I'm thinking I should approach both of these problems in pretty much the same way. I know the following theorem, but I'm not sure where to go from here: "f is continuous at x_0 in dom(f) iff for each epsilon > 0 there exists delta > 0 such that x in dom(f) and |x - x_0| < delta imply |f(x) - f(x_0)| < epsilon.

Any ideas? Thanks!

 

[Dick]

For any delta>0 there is a rational number in the interval |x-x0|<delta. What does that tell you about f(x0)?

 

[HallsofIvy]

Another way to do this is to use the fact that $\lim_{x\rightarrow a} f(x)= \lim_{n\rightarrow \infty} f(x_n)$ where ${x_n}$ is any sequence of numbers converging to a. In particular, there always exist a sequence of rational numbers converging to a.