If I say that $J$ (the volume current density) is proportional to $s$, the distance from the center axis of a long cylindrical wire, I can say that as $s$ changes, so does $J$.
I feel as though I could say the same thing as $f(x)=x^2$. As $x$ changes, $f$ also changes (according to the specified relationship, $x$ raised to the power of $2$).
Could someone help me clarify?
On
Proportionality implies a linear relationship.
A function is much more general. The function $f(x) = x^2$ is quadradic, not linear.
The statement that some quantity $J$ is proportional to another quantity $s$ is equivalent to the statement that there exists some unspecified constant $k$ such that $$ J = ks. $$ This is the same as saying that $J(s)$ is the function $J(s)=ks$, so proportional to implies is a function of. However, a general function $J(s)$ need not satisfy a proportionality relation. For example, $$ J(s) = e^s $$ is an example where $J$ is a function of $s$, yet $J$ is not proportional to $s$. Hence, being proportional to is a strictly stronger statement than being a function of.