A sufficient condition for linearity? If $f$ is a linear function (defined on $\mathbb{R}$), then for each $x$, $f(x) – xf’(x) = f(0)$. Is the converse true? That is, is it true that if $f$ is a differentiable function defined on $\mathbb{R}$ such that for each $x$, $f(x) – xf’(x) = f(0)$, then $f$ is linear?

If $f\in C^2$ (it is twice differentiable, and $f''$ is continuous), then the answer is yes; I don't know if it's necessarily true without this hypothesis.
If $f(x)-xf'(x)=f(0)$ for all $x$, then 
$$f(x)=xf'(x)+f(0),$$
so that
$$f'(x)=f'(x)+xf''(x)$$
$$0=xf''(x)$$
This shows that $f''(x)=0$ for all $x\neq0$, but because $f''$ is continuous this forces $f''(x)=0$ everywhere. Thus $f'$ must be constant, and thus $f$ must be linear.
