Correct option is C
Concept:
If a function is continuous on a dense set S, it doesn't necessarily imply that the function is
continuous on all of R , especially on R∖S, the complement of S in R .
Solution:
Option 1: Continuity on a dense subset S does not imply continuity on the whole set R.
A function can be continuous on a dense subset but exhibit discontinuities on R∖S.
Therefore, this option is incorrect.
Option 2: g is defined only on S, so even if g is continuous on S, it says nothing about f's
continuity on the rest of R. Continuity of g does not guarantee the continuity of feverywhere.
Hence, this option is incorrect.
Option 3: If g(x)=f(x)=0 for all x∈S (which is dense in R), and f is continuous on R∖S,
by the density of S,f must be 0 everywhere on R, 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: g being identically 0 on S and f being continuous on S) does not imply f is identically 0
on R∖S. Continuity on S does not extend to the whole set without further conditions.
Therefore, this option is incorrect.
The correct answer is Option c.
