I have been working on a math problem which I devised myself but am unable to fully solve. It is based on the "additive function" let
f:R-->R such, as f(x+y)=f(x)+f(y). Trivially, the linear function having y-intercept at zero are additive functions, namely, y=ax.
However, this does not PROVE that all real valued additive function must be a linear function of form y=ax.
Thus, I attacked the problem like this, let f be a real valued function having the additive property AND a DIFFRENCIABLE function. Then we have the following f'(x)=lim f(x+h)-f(x)/h and based on our initial hypothesis f(x) is diffrencial thus f"(x) exist everywhere. But, since f is additive we have
f'(x)= lim f(x)+f(h)-f(x)/h thus we are left with lim f(h)/h. But since f is a diffrencialbe function then lim f(h)/h exists futhermore it is the same value regradless of x. Thus, f'(x)=L. Thus, f(x)=ax+b. If b is not zero than by basic observation we could conclude the f is not additive however if b is zero than it is additive. Thus, we have proved a additive function that is diffrenciable must be a linear function of the form y=ax.
Now the problem remains what happens if f is not diffrenciable? Then must it be a linear function of the form y=ax? So I decided to attack the problem as follows. I already worked it out for diffrenciable functions. But what about countinous functions? Must it be that a countinous function having the additive property be a linear funtion of the form y=ax? Try as I might I have not been able to prove it, nor find a counterexample.
And finally, the biggest challenge to prove all additive functions are linear functions. Which I completely doubt that it is true. Interestingly enough I was not able to find a counterexample of a non-linear function. I am guessing there are probably Infinitely many such functions, non of which are elementary.