1. The problem statement, all variables and given/known data Let A be a dense set**. Prove that if f is continuous and f(x) = 0 for all x in A, then f(x) = 0 for all x. **A dense set is defined, in the book, as a set which contains a point in every open interval, such as the set of all irrational or all rational numbers. 2. Relevant equations N/A 3. The attempt at a solution I looked in the answer book, and spivak's proof is much different than mine. I want to know whether mine is adequate. I began by considering a point (α, f(α)) such that f(α)<0. I then stated that there would have to be some σ so that f(x)<0 for all x such that α-σ < x <α+σ. This, however, contradicts the initial statement that A is a dense set (since f<0 for all x in (α-σ, α+σ)). I had an almost identical argument for f>0, which i wish not to type. So does this prove that f(x)=0 for all x?