Sir, this is my doubt please clarify, Let f:R->Rbe a function given by f(x+y) = f(x)+f(y) for all x,y belongs to R such that F91)=a, f(x)= Sir, Kindly give me the solution

Lim h->0 [ f(x+h) - f(x) ] / h = f`(x) Lim h->0 [ f(x) + f(h) - f(x) ] / h = f`(x) As f(x+y) = f(x) + f(y) Lim h->0 f(h) / h = f`(x) Transforming into differential eq. y/x = dy/dx (variable separable) -> log y = log cx at x=1 y=a gives c=a -> log y = log ax Taking antilog both the sides ans: y = ax