Returns to Scale Microeconomics
Seems like the only function $f$ that fits your description
$$
\forall i: \frac{\partial f(\mathbf{x})}{\partial x_i} = c_i
$$
is
$$
f(\mathbf{x}) = A + \sum x_i c_i.
$$
(Frequently $f(\mathbf{0}) = 0$ is assumed. The assumption is referred to as &quot;no free lunch&quot;.)
Then you can apply the definition of returns to scale.