If f ( x ) and g ( x ) are two real valued functions such that f ( g ( x + y ))= f ( g ( x ))+ f ( g ( y )), g (1)=2 and f (2)=1 , then the function g(f(x)) is discontinuous on the set

Step-by-step solution

f ( g ( x + y ))= f ( g ( x ))+ f ( g ( y )) ...(i) and f(2)=1, g(1)=2 Now, from equation (i) : f ( g (2))= f ( g (1+1))= f ( g (1))+ f ( g (1))=1+1=2 aligned & f ( g (3))= f ( g (2+1))= f ( g (2))+ f ( g (1))=2+1=3 & f ( g (1))= f ( g (2-1))= f ( g (2))+ f ( g (-1)) aligned aligned & 1=2+ f ( g (-1)) f ( g (-1))=-1 & f ( g ( n ))= x aligned So, f and g are inverse of each other. Then f(g(x))=g(f(x))=x . g ( f ( x )) is continuous on IR. g ( f ( x ) is discontinuous on