Let S be a dense subset of R and f : ℝ → ℝ a given function. Define g : S → ℝ by g(x) = f(x). Which of the following statements is necessarily true?
1
If f is continuous on the set S, then f is continuous on the set ℝ\ S
2
If g is continuous, then f is continuous on the set S
3
If g is identically 0 and f is continuous on the set ℝ \ S, then f is identically 0
4
If g is identically 0 and f is continuous on the set S, then f is identically 0