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I see this quite commonly, strictly speaking, is $f$ the function and $f(x)$ a number (whose precise value can vary depending on $x$)?

For example I see, sometimes written that 'if $x$ is a variable than $f(x)$ is a function', again if $x$ is a variable it would seem that $f(x)$ is a value that can be seen as varying, and $f$ is a function.

In the strictest of senses if referring to a function $f(x)$ as a function $f$ of $x$ incorrect? As it seems a function should be allowed to be applied to any variable to get a value that depends on it?

If I write $\frac{df}{dx}$ is it best to only ever apply $f$ to $x$? as if I have $f(a)$ finding the derivative of $f$ at $a$ as $\frac{df}{dx}(a)$ is strange as $x$ has nothing to do with $a$?

I have also been told that $f(x)$ is a function as $f(x)$ would be constant, however if $x$ is clearly a variable then $f(x)$ can clearly take different values at different $x$.

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    To be precise we have to use the symbol "$f$" as the name of a function and we have to use e.g. $f(a)$ to name the value (example: a number) of the function for input $a$. – Mauro ALLEGRANZA Oct 13 '22 at 08:54
  • See also the post Why do we use brackets for function parameters? for historical sources. – Mauro ALLEGRANZA Oct 13 '22 at 09:02
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    IMO "if x is a variable than f(x) is a function" is quite meaningless... where did you find it? We have e.g. a function $f$ and we usually write $f(x)$ to specify that the function has one argument, as well as $f(x,y)$ to specify that it has two, and so on. The fact that $f$ is a function derives from its definition and not from the fact that $x$ is a variable. – Mauro ALLEGRANZA Oct 13 '22 at 09:04
  • What Mauro says! $f(x)$ is the value of the function $f$ at $x$, though this value could itself be a function again (this is the case in functional analysis and PDEs, but this is a more advanced topic and a function then is rather called and operator; for example, the operator that assign a smooth function some other function involving its derivative) – MyCatsHat Oct 13 '22 at 09:08
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    As Mauro says, $f$ is the function (e.g. $x^2$ over the integers), $f(x)$ is its evaluation (e.g., $f(x)=4$ at $x=2$). In other words, $f$ is the mapping itself, while $f(x)$ is one number (assuming that's what we get out). But using $f(x)$ to sloppily denote the function is so common that the rule doesn't apply very strictly in practice. – Jam Oct 13 '22 at 09:11
  • @jam $f(x)$ does not necessarily have to be a constant at $x$ if $x$ represents any point of the domain $f(x)$ is a value which depends on $x$? –  Oct 13 '22 at 09:12
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    "f(x) is a value which depends on x?" Yes, we parse the notation as meaning that $f(x)$ is a function of one argument: when we choose a value $a$ (from the domain) for the argument we can use the definition (formula) of the function to "compute" the value $f(a)$ of the function. – Mauro ALLEGRANZA Oct 13 '22 at 09:15
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    The point is that this is an abuse of notation. Strictly speaking, none of it is a function since domain or codomain are nowhere specified. If you really want to be precise, write it like this: "let $f\colon X\to Y$ be a function given by formula $f(x) = \text{bla bla bla}$". – Ennar Oct 13 '22 at 09:16
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    Having said that - that is the "common understanding" of an introductory level course in modern mathematics - what do you mean with "f(x) does not necessarily have to be a constant at x if x represents any point of the domain"? Obviously not all functions are constant one... – Mauro ALLEGRANZA Oct 13 '22 at 09:16
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    Discussed already here, here and here. – Mauro ALLEGRANZA Oct 13 '22 at 09:22
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    @user1007028 That depends on what you mean by a constant/variable, really. If $x$ is "realized" as a particular number (i.e. is set to a certain value), then $f(x)$ will be realized too, e.g. if $x$ is realized as $2$, then $f(x)$ will be realized as $f(2)=\ldots$. On the other hand, if $x$ is set as a variable, then $f(x)$ will be a variable too, but it will simply be an un-realized form of the same type of object as $f(2)$ in the previous case. That is, $f(x)$ would still be a number, just one corresponding to a yet undetermined argument. – Jam Oct 13 '22 at 09:26
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    To draw an analogy that might help, a function is like a lookup table with two columns (arguments on the left, values on the right) and many rows, each row for a particular argument. By picking any row, we can see an argument and its value. One row may look like $|2|4|$ and another like $|5|25|$. In the general case, a row will look like $| x | f(x) |$; this is the meaning of $f(x)$, the right-hand side of one row. In contrast, $f$ expresses the whole table, i.e. the mapping collectively. – Jam Oct 13 '22 at 09:33
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    See also the post In written mathematics, is f(x)a function or a number? as well as Why does notation for functions seem to be abused and ambiguous? – Mauro ALLEGRANZA Oct 13 '22 at 09:37
  • @Jam why is that people try to say $f$ is a function of $x$ and associate with a particular variable when we quite literally can apply $f$ to any variable or number we wish? –  Oct 13 '22 at 10:12
  • @user1007028 Because sometimes you want to talk about the function as a whole, but other times you want to talk about the values it's taking :) Using the above "table" analogy, we might wish to say that $f$ is increasing, which is a statement about the whole table (i.e. the function). But we might wish to say that there is an $f(x)$ equal to $4$, which is a statement about one particular cell in a row (i.e. a value). – Jam Oct 13 '22 at 10:16
  • @Jam that makes sense but my only doubt is why call it a 'function of $x$' the function is the function no matter what variable you put into it's input, it doesn't care what variable you use. –  Oct 13 '22 at 13:08

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There is a lot of sloppyness going on, that’s for sure. Most of the times people will understand what is meant. If $f$ is a function and $x$ is a variable $f(x)$ would indeed denote an evaluation of $f$ in $x$, although that does not necessarily make sense, since $x$ is a variable. So one could interpret this as $x$ being a placeholder for a value and $f(x)$ being a placeholder for an evaluation.

Of course one could write more correctly $f$ or $x\mapsto f(x)$. In terms of derivatives (assuming that $f$ is in fact differentiable in each point) $\frac{\mathrm d f}{\mathrm d x}$ would be a function that maps a point to the derivative in that point. So indeed $$\frac{\mathrm d f}{\mathrm d x}(a)$$ is a correct spelling. $x$ is in this case just a symbol to define a specific argument of the function. You can think of this somewhat like $$\frac{\mathrm d (x\mapsto f(x))}{\mathrm d x}(a)$$

Then we have the question: Is a constant a function? That depends on what your model is. How do you define sets and how do you define functions? A classic way to define a function would be a subset of the cartesian product between two sets that satisfies certain conditions. In this sense a constant is not the same as a constant function (which would be $A\times\{a\}$ for some domain $A$ and some value $a$).

But of course any value can be interpreted as constant function. We do this all the time, or more general we interpret symbolic expressions as functions. For example if we say $$ \int_0^1 1 + \sin(x)\,\mathrm dx$$ we actually mean $$ \int_0^1 (x\mapsto 1+\sin(x))\,\mathrm dx$$ If we say $$ \int_0^1\int_0^1 f(x,y)\,\mathrm d x\,\mathrm d y $$ we actually mean $$ \int_0^1(y\mapsto\int_0^1 (x\mapsto f(x,y))\,\mathrm d x)\,\mathrm d y $$

Again, it is general sloppyness that is okay, because everyone knows how this is to be read.

Lazy
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    "There is a lot of sloppyness going on" maybe... but language, and also "mathematical jargon", is made for COMMUNICATION. Do you really think that someone during a classroom teaching will agree on the usefulness of writing on the blackboard the last formula above instead of the "usual" one? – Mauro ALLEGRANZA Oct 13 '22 at 09:54
  • @MauroALLEGRANZA I’m not claiming that this sloppyness or informalism is bad. I’m clearly stating "Again, it is general sloppyness that is okay, because everyone knows how this is to be read." – Lazy Oct 13 '22 at 09:55
  • @Lazy I think in the case of an integral writing $f(x)$ is alright as you can see it as the reimann sum of the height of the graph (value of $f$ at $x$) and a small change in $x$ ($dx$), when you say that $f$ at $x$ doesn't make sense, why? It makes sense to me $x$ is an arbitrary element and $f(x)$ is an also an arbitrary element –  Oct 13 '22 at 10:06
  • @user1007028 As a placeholder $f(x)$ would be an arbitrary but fixed value, which makes it different from the function $f$. Writing $f(x)$ is alright as long as it is clear what is meant. But it is still informal. Also I’m saying $f(x)$ does not necessarily make sense, it depends on how you see it. If you see $x$ as a metatheoretic symbol that can be assigned any value then $f(x)$ can be seen as a symbol that stands for the value of $f$ evaluated at the value assigned to $x$. – Lazy Oct 14 '22 at 11:03
  • @user1007028 If instead we see symbols as mathematical objects then we have the problem that $x$ is not in the domain of $f$, so $f(x)$ does not make sense. – Lazy Oct 14 '22 at 11:03
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For example I see, sometimes written that 'if x is a variable than f(x) is a function', again if x is a variable it would seem that f(x) is a value that can be seen as varying, and f is a function.

if x is a variable, then f(x) is a function

The above statement is not correct. x being a variable does not imply f(x) to be a function.

Definition

A function or (sometimes called a map) f from a set A into a set B is a rule that assigns a value f(a) belonging to B to each element "a" belonging to A. The input set A is called the domain of the function.

Note that f is the function, while f(x) is the value assigned by the function at the element x that belongs to A.

Source: Elementary Real Analysis By Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner

In the strictest of senses if referring to a function f(x) as a function f of x incorrect?

According to the above definition, and my personal interpretation of it, to call f(x) a function, is incorrect for the reader who reads the sentence literally but it is valid as a shorthand convention within the context of the contemporary mathematics literature.

The name "x" (or "a") is an arbitrary name representing a specific instance of the elements in the domain set A and whatever you choose does not affect the mechanism of the mapping. The domain set A can contain any number of elements such as A = {1,2,3,6} any one of theses elements can be represented by a or x in the expression denoting the function value f(x) as long as x belongs to the set A.

I have also been told that f(x) is a function as f(x) would be constant, however if x is clearly a variable then f(x) can clearly take different values at different x.

When the function maps each element of the domain set to a single value, then we say that the function is a constant. so f(1)=f(2)=f(3)..., when we use the letter "x" in the expression f(x) we mean the value of the function f for any value, "x".

I hope this clarifies your inquiry some...:)

Reference Link

NoChance
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