I am given the assumption that a function $f$ has the property that $f(x+y)=f(x)+f(y)$. I am wondering if it follows that $f(x-y)=f(x)-f(y)$. I think that this is false, yet I see this being used in proofs.
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Of course it's true. f(x-y)+f (y)=f (x). It's also true that f (q)=qf (1) for all rational q. – fleablood Apr 21 '18 at 06:01
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Your condition is the definition of $f$ being a group homomorphism for addition. Your question is the sort of thing that would be treated as an exercise in chapter one of a group theory textbook: for a group homomorphism $f$, we always have $f(-x)=-f(x)$ and $f(0)=0$. – Jack M Apr 21 '18 at 10:26
