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I often see notation/wording like "let $f(x)$ be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$".
I would say that $\sin$ and $x \mapsto \sin(x)$ are functions, while $\sin(x)$ is a real number.
Are there any correct or best practices in this regard?
Saying that $f$ or $x\mapsto f(x)$ is the function rather than $f(x)$ is the more common modern perspective in higher mathematics. I have even seen a precalculus book that stresses that $f$ is a function, whereas $f(x)$ is a value of the function. The notation $f(x)$ to denote a function remains because it is often more convenient, and it is especially prominent from high school mathematics up to calculus because it is psychologically easier to become accustomed to. This may lead to ambiguity, because in any case $f(x)$ sometimes does refer to a value of the function, but the strictest logical correctness of notation isn't for everyone, or for all situations.
I find it off-putting when I read mathematics that was written many decades ago and come across something like, "Let $\varphi(x)$ be a bounded linear functional on $X$....". I think to myself, "No, $\varphi$ is the functional!" Then I take a deep breath and relax, and find that this notation did not hinder the authors from doing and writing great mathematics.
Ahlfors's complex analysis text has a footnote on page 21 of the 2nd edition, 1966:
Modern students are well aware that $f$ stands for the function and $f(z)$ for a value of the function. However, analysts are traditionally minded and continue to speak of "the function $f(z)$."
You are right. Properly speaking, the function is called just $f$, and its value at the point $x$ is denoted by $f(x)$. Speaking of "the function $f(x)$" is what mathematicians call "abuse of notation", but it is of course often very practical.
(By the way, one usually writes $x \mapsto \sin x$, not $x \to \sin x$; it's "\mapsto" in TeX.)
You are quite right: $f$ by itself should denote a function, $f(x)$ by itself should denote the element in the codomain of $f$ (in this case, the real number) that results when you evaluate $f$ on the element $x$ of its domain, where $x$ should previously have been defined.
However, this rule is honored as much in the breach as the observance: there are many situations where it is convenient to break it. When defining a function by a formula, it's hard to avoid a dummy variable, and so one likes to say "let $f(x)=e^{-x}$" instead of the more correct but awkward "let $f$ be the function $x \mapsto e^{-x}$". In particular, with functions that have multi-letter symbols like $\sin$, I find that people generally prefer to avoid writing them without an argument like $\sin x$. One does not like to talk about $\sin$ as a function in itself, so instead of writing something like $\sin'' = -\sin$, one would rather say "if $f(x)=\sin x$, then $f'' = -f$".
An alternative to a dummy variable that's sometimes used is a dot: $\cdot$. People sometimes write "let $f=g(\cdot + 5)$" to avoid the less correct "let $f(x)=g(x+5)$".
When working with functions of several variables, using dummy variables often helps keep track of which variable is which. One often writes something like: "let $u(x,t)$ be a solution of the heat equation $\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0$". Of course, one is really talking about the function $u$ and not any particular real number of the form $u(x,t)$, but it would be much more awkward to write otherwise. It also reminds you that the first argument of $u$ should be interepreted as space and the second one as time.
In short: mathematicians are not compilers. Written mathematics has some syntax rules, but they are not quite hard-and-fast, and need not be followed at the expense of clarity.
let f x = exp (-x) is just syntactic sugar for let f = fun x -> exp (-x) in ML, for instance. In particular, it is perfectly consistent with the meanings of f and f x and does not break any syntax rules.
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The wording "let $f(x) \in C^0(\mathbb{R})$..." is really sloppy. Of course it doesn't lead to confusion, but there is something to be said for correctness. Ideally whenever you see $f(x)$ it should be because we care about what $f$'s values are rather than $f$ in and of itself as say a member of some space of functions.
Oh, and your example of "$x \to sin(x)$" is IMO better written as "$\mathbb{R} \owns x \mapsto \sin x \in \mathbb{R}$. Obviously not so useful for this particular function, but very smart for things like "$V \times V \owns (x,y) \mapsto \langle x,y \rangle \in \mathbb{C}$".
Strictly speaking, I would say that $f(x)$ is the value of the function $f$ evaluated at $x$. However, $\frac{x^3-2}{x+1}$ might be used as a function; it is probably because $x\mapsto\frac{x^3-2}{x+1}$ is harder to write and takes up more space.
It all depends on what you have said $x$ is.
If $x$ is previously defined as a number (e.g. $x = 3$) then $f(x)$ is also a number, since you can evaluate $f$ at a certain point $x$ to get the number. (A more common way to notate that would be to say $x_1$ instead of $x$...)
If, on the other hand, $x$ is defined as part of some set (e.g. $x\in[0,1]$) then $f$ is a function defined for input values in that interval. $\sin x$ is defined as a function for $-\infty < x < \infty$, but as a number for $x=\pi/2$.
