Let's consider a function f: R -> R and a subset S ⊂ R. If for each sequence {x_n} ⊂ S which converges to some limit 'a' in S, the sequence {f(x_n)} also converges and the limit of {f(x_n)} equals f(a), which of the following statements is true?

 

A) The function f is continuous on the subset S.

B) The function f is discontinuous on the subset S.

C) The function f is continuous on the entire real number line.

D) The function f is uniformly continuous.

E) The function f is continuous only at the point 'a'.