Why doesn't superposition imply linearity? Why is homogeneity needed?

Question

If I have a function which satisfies superposition I know $f(x_1+x_2)=f(x_1)+f(x_2)$. If I had now an element making f inhomogeneous ($f(0)\neq0$) this element would occur once on the left hand side and twice on the right hand side. Thus superposition would also not hold. I don't understand why superposition doesn't imply linearity. Which functions satisfy superposition but not homogeneity?

Answer 1

Over any vector space over $\mathbb R$, all continuous functions satisfying $f(x_1+x_2)=f(x_1)+f(x_2)$ also satisfy $f(\alpha x)=\alpha f(x)$ - so continuity and superposition implies linearity when we work with the reals.

However, this is not true of other fields; firstly, "continuity" might be harder to define in other fields (or in infinitely many dimensions), which is itself a problem. However, even over fields like $\mathbb C$ where there is a clear notion of continuity, we end up with functions like: $$f(a+bi)=a$$ which satisfy superposition but are not homogeneous, since $f(i\cdot 1)=0$ which is distinct from $i\cdot f(1)=i$.

Answer 2

$0+0=0$, so $f(0) = f(0)+f(0)$ which implies $f(0) = 0$. Thus, all functions which satisfy "superposition" also satisfy "homogeneity".