U don't need at all that mumbo-jumbo ( concavity -convexity or the above proof ) to solve the functional eqn

It is very simple to solve .

first differentiate partially both sides wrt 'y' , yielding

 ....................(1)

Now put y=0 in both sides , yielding f'(x/2)=f'(0) = constant .......(2)

from (2) it follows that f(x) =mx +c ( proved)