Function and Image Element

This section is fairly important — it introduces some important concepts in … functions. Alright, we just want to tell you that the function and its image elements are not the same thing. Well, you may find this ostensible. However, the denotations of function and its image elements can be really misleading. You may clearly identify that something like \(f\) and \(f(x_0)\) are different things. But can you always notice that \(f\) and \(f(x)\) are different things, especially when \(x\) is some variable instead of constant like \(x_0\)?

There are no intentions to detailedly discuss the issue, since it will finally becomes a philosophical war between objective and formal Mathematics. Here, we directly take the view of objective Mathematics. Under this philosophy, the function \(f\) is a mathematical object which is a subset of the Cartesian product of domain and image (satisfying some conditions); while \(f(x)\) is an element of image (no matter whether \(x\) is a constant or not). Therefore, they are essentially different objects.

To help understand this, especially the differences it causes in Quantum Field Theory, we set up the following examples

Example 1. In Lagrangian formalism and Path Integral Formalism respectively, two kinds of functional derivatives are invoked — \(\delta/\delta f\) and \(\delta/\delta f(x)\). The former is the derivative with respect to the function \(f\), and the latter is the derivative with respect to the element \(f(x)\). You will see in that it is the definition of the latter one which causes the famous infinity problem in Quantum Field Theory; while nothing happens to the former one.

Example 2. In some cases, we will write something like


Although there is a equality sign in the equation, there are essential difference between \(f_x\) and \(f\) — the former is a family of functions of \(y\) (labelled by \(x\)) and the latter is a function of two variable \(x\) and \(y\). The equality essentially means solely that the value is the same.