OK, I think I have a proof of the following:
If , then for some real m.
Setting and , we have ;
in other words .
Next, set and , which yields .
Since , we have , or
Let (the left side)
Then we see that and is an odd function, i.e. .
Setting equal the right sides of the above two equations and simplifying,
which is the definition of linear function with . (y-intercept = 0)