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? 

Concept:

If a function is continuous on a dense set , it doesn't necessarily imply that the function is

continuous on all of  , especially on , the complement of   in  .

Explanation:

Option 1: Continuity on a dense subset  does not imply continuity on the whole set .

A function can be continuous on a dense subset but exhibit discontinuities on .

Therefore, this option is incorrect.

Option 2:  is defined only on , so even if  is continuous on , it says nothing about 's

continuity on the rest of . Continuity of  does not guarantee the continuity of  everywhere.

Hence, this option is incorrect.

Option 3:  If  for all  (which is dense in ), and  is continuous on ,

by the density of ,  must be 0 everywhere on , because a continuous function on a dense set that is

0 must be 0 on the entire set. Therefore, this option is correct.

Option 4:  being identically 0 on  and  being continuous on ) does not imply  is identically 0

on . Continuity on  does not extend to the whole set without further conditions.

Therefore, this option is incorrect.

The correct answer is Option 3).