Lets us take an example: the funktion $f(x)=x^2$. I how realised that this notation does not make much sense. But how would one write it then?
How about the following:
$$f:x\mapsto x^2$$
I'm worried that it will be confused with $f:A \longrightarrow B$, where $A$ is the domain and $B$ is the codomain.
Can you guys suggest a "better" notation?
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$\begingroup$When you write $f(x) = x^{2}$, you are saying that the application of $f$ at an element $x$ is given by $x^{2}$; in other words, you are evaluating $f$ at some $x$. When the function depends on one variable only, we usually use this as a representation of the map which is defined by $f$ because $x$ is assumed to be arbitrary and $x$ is a dumb variable. When we consider functions of two variables, for instance, this representation sometimes has to be enhanced when one wants to make explicit the difference of an application of $f$ at an arbitrary point and $f$ as a function which depends on some variable. This is why, sometimes, we write $f(x,y)$ as $f(\cdot, y)$; this means that $f$ is fixed at some arbitrary value of $y$ and is a function of $x$ only. The notation $x \mapsto x^{2}$ is a more direct one, and it captures the essence of the map but it does not make the domain or range explicit, so this is more like a quick notation. To me, the more complete representation of a function whose domain is $A$ and the range is $B$ is written as $$\begin{align} f: A &\to B \\ x &\to f(x) = x^{2}\end{align}$$
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