The correct options are
A f(x)=mx+n, n∈Z
C f(x)=0
D f(1),f(4),f(7),f(10) are in A.P.
mf(x)+f(my)=f(f(x+y)) ...(1)
At y=0⇒mf(x)+f(0)=f(f(x)) ...(2)
Replace x by x+1 in eqn(2)
mf(x+1)+f(0)=f(f(x+1)) ...(3)
Now, put y=1 in eqn(1), we get
mf(x)+f(m)=f(f(x+1)) ...(4)
From eqn(3) and (4), we have
mf(x+1)+f(0)=mf(x)+f(m)
⇒f(x+1)−f(x)=f(m)−f(0)m=Constant
Since, the difference of two consecutive terms f(x+1) and f(x) is constant.
Therefore, f(1),f(4),f(7),f(10) are in A.P.
Let f(x)=kx+n ; n∈Z
From eqn(1),
m(kx+n)+kmy+n=f[k(x+y)+n]⇒km(x+y)+mn+n=k2(x+y)+kn+n
Comparing the coefficients of (x+y) and constant term, we get,
km=k2 and mn+n=kn+n
⇒k=0,m
When, k=0⇒mn+n=n⇒n=0
When, k=m⇒mn+n=mn+n⇒n=n
⇒n can be any value.
When k=0, n=0⇒f(x)=0,
When k=m, n∈Z⇒f(x)=mx+n