1. The problem statement, all variables and given/known data If f is continuous, and f(x) = 0 for all x in A, where A is a dense set. Then f(x) = 0 for all x. I am using the following definitions: A set of real numbers A is dense if every open interval contains a point of A. And the limit definition for a continuous function. 2. Relevant equations 3. The attempt at a solution Suppose there is an a such that f(a) > 0, then since f is continuous, there is an δ > 0, such that, for all x, if |x - a| < δ, then |f(x) - f(a)| < f(a). So f(x) > 0 for all x in (-δ + a, δ + a). But since (-δ + a, δ + a) is an open interval it contains a point of A call it z, but that is impossible, because then f(z) = 0 (By hypothesis), and f(z) > 0 because f(x) > 0 for all x in (-δ + a, δ + a). Assuming there is an a such that f(a) > 0 leads to a contradiction. I did basically the same to show that there are no x with f(x) < 0. Therefore f(x) = 0 for all x. Am I correct? Also, hello.