f+g may be a continuous function if

Consider h(x)=f(x)+g(x) Iff(x) is continuous and  g(x) is discontinuous , then let us assume that h(x) is continuous.Now,       g(x)=h(x)︸continuous  function−f(x)︸continuous  function ⇒         g(x) is a continuous function.Which is contradictory to the given fact that g(x) is a discontinuous function.Hence our assumption that h(x) is continuous is wrong.i.e., if f is continuous and g is discontinuous then f+g cannot be a continuous function.i.e., (a) wrong.Similarly (b) is wrong.From (c), we takef(x)=x−[x]            (Discontinuous  function) g(x)=x+[x]            (Discontinuous  function) (f+g)(x)=(x−[x])+(x+[x])                 =2x            (Continuous  function)