Although I have not quite studied functional equations, I came upon Cauchy functional equation and tried to prove it. Here is what I have done:
We are given the condition, $f(x+y)=f(x)+f(y)$.
So, for some constant $a$ and another constant $f(a)=b$, we have $f(x+a)=f(x)+b$.
Differentiating both sides wrt $x$, we have $f'(x+a)=f'(x)$.
But this result is valid for any constant $a$, and hence $f'(x)=c$, for some constant $c$. This gives us $f(x)=cx+d$. Putting this result into original condition, we have $c(x+y)+d = cx +d+ cy +d$. Hence $d=0$ and $f(x)=cx$, for some constant $c$.
Is my proof right or are there holes which needs to be filled? I asked here because it is different from the proof I found. My main concern is that I have assumed that the function is differentiable. Is there any elementary way to patch up for non differentiable functions? What about some other points?