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Formally, if $f$ is a function, $f(x)$ is a value. So for instance, $f$ can be continuous, but not $f(x)$. In teaching at school and university, notation is quite often mixed up, e.g. the function is called $f(x)$. This may be problematic but can also have benefits. What good reasons for the use of the different notations do you know?

This question raised from the discussion of this discussion on notation of derivatives.

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  • $\begingroup$ Those interested in these issues may want to read about Karl Menger's attempt to reform calculus notation in the 1940s and 1950s. For more about this, see my comments in this 29 October 2006 post of mine at Math Forum. $\endgroup$ – Dave L Renfro Apr 17 '14 at 17:32
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    $\begingroup$ "Formally, if $f$ is a function,...", actually this way of using the word function is surprisingly modern, and became standard only after ${\sim}1930$. Prior to that, all mathematicians officially called $f(x)$ the function. So that's historically the correct way. See hsm.stackexchange.com/q/6104/3462 $\endgroup$ – Michael Bächtold Apr 19 '18 at 19:41
  • $\begingroup$ For students at the middle school through introductory undergraduate levels, I would insist on very clearly distinguishing between $f$ and $f(x)$, because their understanding of this important distinction is weak/non-existent. At higher levels, one may dispense with this distinction. // The analogy is to teaching young children how to write -- we insist that they dot their i's and cross their t's and penalize them for failure to do so. But by the time they are 12, we may be a little more relaxed about the matter. $\endgroup$ – Kenny LJ Oct 23 '18 at 0:56
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To distinguish not too strictly between $f$ and $f(x)$ allows to operate more easily with functions built up from other functions.

For example, one might want to say things like:

Let us consider the function $\sin (3 x^2)$ on the intervall $[0,1]$.

This can be considered as sloppy in a formal sense, but I think it is still clear and in some contexts preferable over more correct alternatives, in my opinion.

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    $\begingroup$ This answers (intentionally) only covers one aspect, as I think the question is relatively broad and thus "one idea per answer" might be a good format to answer it. $\endgroup$ – quid Mar 21 '14 at 10:07
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    $\begingroup$ I'm used to writing ‘the function $x \mapsto \sin(3x^2)$’. It is necessary when you consider parametric integrals, for example. $\endgroup$ – Jill-Jênn Vie Mar 21 '14 at 11:11
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    $\begingroup$ @Jill-JênnVie I agree one could write it like this, and indeed I had that particular alternative, which I think is one of the better ones, typed out in a draft of the post (but removed this part). Still, I think in some conexts this is neither necessary nor necessarily a good idea. $\endgroup$ – quid Mar 21 '14 at 11:30
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    $\begingroup$ In view of my related question, here is a way of saying the same thing without abuse of notation "Let us consider $\sin(3x^2)$ as a function of $x$ in the region $0\leq x \leq 1$." I don't think this is less clear. $\endgroup$ – Michael Bächtold Sep 29 '16 at 9:14
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To my mind, this abuse of notation is more similar to how I think when I'm doing physics. Imagine a bottle of gas. It has a temperature $T$, a volume $V$, a pressure $P$, a number of moles $N$ and an entropy $S$. They are related by various relations such as the ideal gas law. Which one is a function of which? Well, any one can be a function of any other, under varying circumstances. Insisting that I am going to write everything as a function of temperature, or everything as a function of volume, or whatever, only introduces confusion.

I also tend to think in terms of many related quantities, no one a function of any other, when doing algebraic geometry (my research area) and differential geometry, but those aren't relevant to students in an ordinary calculus course.

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  • $\begingroup$ Your analogy is flawed. In classical physics we always have every variable being a function of time. The ideal gas law or any other physics equation would then simply be a functional equation. In general, related quantities are always functions of some underlying parameters. $\endgroup$ – user21820 Jul 24 '15 at 11:48
  • $\begingroup$ @user21820 : I don't think that you can so blithely assert that everything is a function of time. For example, the idea that the temperature is the partial derivative of the internal energy with respect to the entropy (with volume fixed) is useful even if you are considering a static gas whose energy and entropy (and hence temperature) never change with time, or conversely if you are considering a gas whose volume does in fact change with time (so that we cannot find the relevant partial derivative by performing a calculation with actual values of the energy and entropy at various times). $\endgroup$ – Toby Bartels Sep 23 '18 at 7:08
  • $\begingroup$ You're quite right that all of these quantities can be viewed as functions, but they're not functions defined on the time line (the real line parametrized by time) but rather functions defined on phase space (the abstract space of macroscopic states of the ideal gas). Since phase space can be parametrized by various combinations of these very variables ($P$ and $T$, $P$ and $V$, $S$ and $V$, etc), David is pondering which ones should be the ‘independent variables’. The answer is that there is no need to choose, any more than one must choose time $t$; the phase space itself is the domain. $\endgroup$ – Toby Bartels Sep 23 '18 at 7:14
  • $\begingroup$ @TobyBartels: Where did I ever say that other parametrizations are not useful? Rather, I am stating that, in classical physics, ultimately every concrete variable is a quantity that varies over time, simply because everything happens along the time line. You are wrong to say that the phase space is sufficient, because a closed loop with hysteresis must be parametrized in terms of time rather than any of the variables that underlie the chosen phase space. And I also catered for non-classical phenomena, when I said that in general there are underlying parameters (that may not be time). $\endgroup$ – user21820 Sep 23 '18 at 11:49
  • $\begingroup$ I gave an example where things are not varying in time. Treating all quantities as functions on phase space works for David Speyer's answer, but you're right that it doesn't work for everything. Neither does treating everything as a function of time. $\endgroup$ – Toby Bartels Oct 2 '18 at 3:19
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Several arguable points where made in previous answers, mostly that the abusive notation $x+\sin x$ for $x\mapsto x+\sin x$ is in many circumstances lighter, while the more formal notation avoids some confusions that are common among students.

In my opinion, the most important point is to adapt the notations and the rigor to the kind of students we deal with. In front of first and second year students, I try to always distinguish functions from values (or from expressions with free variables), because the very notion of function is shaky in their minds. I ask them to write things rigorously too, until they master them. With higher level math majors, I start to accept more sloppiness, as seems convenient, and sometimes reminding students the abuse made to notations. In a graduate course in differential geometry, abuses would probably be everywhere (but I did not teach such a course yet).

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  • $\begingroup$ often the language $f=f(x,y)$ is useful to get across the idea that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$. I think this is an example of your method, which I endorse. $\endgroup$ – James S. Cook Apr 20 '14 at 0:47
  • $\begingroup$ @JamesS.Cook: I agree. In PDE, I often wrote like this: Suppose the function $u=u(x,t)$ solves the heat equation $u_t(x,t)=a^2u_{xx}(x,t)$ with some initial/boundary conditions. If I write formally, it seems like: Let function $u: (x,t)\in [a,b]\times [0,+\infty)\mapsto u(x,t)$ solves .... But it is very cumbersome to do so. $\endgroup$ – azc Oct 8 '18 at 8:00
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Perhaps it is worth mentioning here there is another notation which at first glance seems non-rigorous. For example, $f = x^2+\sin(y)+z$. Here $f: \mathbb{R}^3 \rightarrow \mathbb{R}$ is a function built from the coordinate functions $x,y,z$ each of which is itself a function defined in the obvious manner: $$ x(a,b,c)=a, \qquad y(a,b,c)=b \qquad z(a,b,c) = c. $$ At first I resisted this notation, but as the semester wears on I grow to enjoy its simplicity. I encountered in in Barrett O'neill's Elementary Differential Geometry. There is always a tension for me in teaching manifold theoretic topics, it's hard to strike the right balance of notation; too much kills intuition, too little makes calculations seem impossible.

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    $\begingroup$ This notation is in particular a good way to introduce the differential forms $dx$, $dy$, etc. $\endgroup$ – Benoît Kloeckner Apr 19 '14 at 20:35
  • $\begingroup$ This point of view was advocated by Thurston (not that Thurston) in a Monthly article, "What is wrong with the definition of $\textrm dy/\textrm dx$?": jstor.org/stable/2975132 . $\endgroup$ – LSpice Nov 24 '14 at 0:45
  • $\begingroup$ @LSpice thanks. I'll have to read that when I get back to school... $\endgroup$ – James S. Cook Nov 26 '14 at 13:56
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Consider the following observations:

  • People shorten things with increasing frequency of usage. For example, the most frequent words are short. This is a way of making communication more effective, e.g. see Huffman coding. For example this list has its first 5-letter word at 39th place (first three are for comparison): $$\begin{array}{c|c|c|c} \textbf{place} & \textbf{word} & \textbf{letters} & \textbf{count} \\\hline 1&\text{you}&3&1222421\\ 2&\text{I}&1&1052546\\ 3&\text{to}&2&823661\\ 39 & \text{about}&5 & 142750 \\ 70 & \text{really} & 6 & 68311 \\ 93 & \text{something} & 9 & 54736 \\ 95 & \text{because} & 7 & 53566\end{array}$$
  • If there is only one free variable in expression $\mathcal{E}$, saying "function $\mathcal{E}$" is unambiguous. The key word here is "function" which let us know that what follows should be treated as a function, and not just as an expression (with appropriate context it might be not necessary). Such formula might be unambiguous even with multiple free variables, if the convention designates one as such (e.g. $x$ versus $a$), compare \begin{align} &\text{function } &x^2&+\sin x &\text{ok}\\ &\text{function } &ax^2&+\sin bx&\text{ok}\\ &\text{function } &c^2&+\sin bd&\text{ambiguous} \\ &\text{function } &d\mapsto c^2&+\sin bd&\text{ok} \end{align}
  • Human thinking is fuzzy. Our brains do not process mathematical concepts formally (even if we are capable of it). There is a whole thread about notation abuse at math.se, in particular it includes a great quote by Gila Hanna: \begin{align}\text{The student of mathematics has to develop a tolerance}\\\text{for ambiguity. Pedantry can be the enemy of insight. }\end{align}

I suspect that there are also multiple other reasons, but the above seem to be the most important. That being said, I think it is alright to tolerate such abuse in speech (which is ambiguous and ill-suited for math anyway) or if time constraint is severe. However, I would not accept such a style in homework assignments. Moreover, I urge the teachers to write correctly on the board during their classes, it's not a significant burden and it might help, e.g. to avoid confusion.

I hope this helps $\ddot\smile$

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It is interesting to see how computer algebra systems deal with this kind of thing. In Maple, for example, you can do the following:

  • Define f := x^3
  • Enter diff(f,x) to get the derivative
  • Enter subs(x=3,f) to evaluate.

Alternatively:

  • Define g := (x) -> x^3
  • Enter D(g) to get the derivative
  • Enter g(3) to evaluate

You can convert between the two idioms by f := g(x) and g := unapply(f,x).

When using Maple for research I find myself mixing the two idioms according to what is convenient for the immediate purpose. This feels faintly unsatisfactory, but it works. I used to teach a Maple-based undergraduate course, which also used both idioms. The difference between them did cause some confusion, but it was not especially prominent compared to other things that the students found difficult.

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    $\begingroup$ It's similar in Mathematica, where you can define $\mathtt{f[x_-]:=x^3}$ or $\mathtt{f := (\#^3)\&}$. $\endgroup$ – user173 Mar 21 '14 at 12:59
  • $\begingroup$ In maxima you'd define f(x) := x^3, after which f is just a name (no meaning attached). $\endgroup$ – vonbrand Mar 21 '14 at 13:22
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    $\begingroup$ There's no abuse of notation here, because f is not a function; g is, and requires appropriate handling. $\endgroup$ – dtldarek Mar 21 '14 at 14:12
  • $\begingroup$ Specifically, f is an expression (a string of symbols that may contain free variables) while g is a function (an object that takes an input and produces an output). diff takes an expression and a variable name as inputs, but D is a (higher-order) function. They are not idioms for the same thing at all! If students are confused it is because they were not taught the difference between expressions and functions... $\endgroup$ – user21820 Jul 24 '15 at 11:38
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It was mentioned in other answers that having a more sloppy notation is better to not complicate the communication. This is okay for people who had really understood the concepts of mathematics.

I have the feeling that understanding what a function is one of the those things students don't really understand. Common mistakes (and in my opinion all of them are led by the fact of the sloppy notation) are:

  • "A function is something you can write down". For example, most students have a lot of problems if you define a function in a more indirect way: For every argument, the function takes as a value the minimizer of some functional (dependent on the argument). In abstract algebra or differential geometry there are also very sophisticated examples of a very abstract usage of functions.
  • By all the time taking $x$ as argument, some students are not able to think about another variable as argument of a function which is the case in many applications.
  • The lack of not distinguishing between function and its evaluation leads often to a lack of the ability to use theorems, as it was also pointed in this question about the chain rule: Revisiting topics from previous courses.
  • If students fail to understand the deep meaning what a function is, there is no chance that they can understand basic concepts in e.g. functional analysis, where "values" are functions.
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    $\begingroup$ The second point is so true. I am teaching a second year engineering class in differential geometry, and in class the oriented curvature for a curve $c$ parametrised by $t$ is written as $\kappa(c,t)$. The students got immediately lost when on a homework question they have to treat the curve $s\mapsto \beta(s)$. $\endgroup$ – Willie Wong Mar 21 '14 at 10:31
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    $\begingroup$ Yours is the only proper answer... It's disappointing that the other answers are more highly valued... (Though dtldarek did make clear in his answer that he prefers to be precise in writing, he didn't address the underlying issue of why we should have such precision.) $\endgroup$ – user21820 Jul 24 '15 at 11:49
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In most contexts, it makes sense to say that

"$x + \sin(x)$ is a continuous function".

Consider the alternatives:

"the function taking $x$ to $x+\sin(x)$ is a continuous function"

"$\text{id} + \sin$ is a continuous function"

Take your pick.

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    $\begingroup$ I would say "function $x+\sin x$ is continuous", but would write "$x\mapsto x + \sin x$" instead. Speech is ambiguous and ill-suited for mathematics anyway, but it's definitely not a hindrance to make it correct in writing (and perhaps enough for the students see and understand the distinction). $\endgroup$ – dtldarek Mar 21 '14 at 10:35
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    $\begingroup$ Or perhaps, "the function $f$ such that $f(x) = x + \sin(x)$ is continuous". $\endgroup$ – DavidButlerUofA Jul 2 '14 at 10:28
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    $\begingroup$ My pick is your #2, that is, "The function $x\mapsto x+\sin x$ is continuous". $\endgroup$ – Did Nov 16 '14 at 16:52
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    $\begingroup$ I always go for ‘$\mathrm{id} + \sin$ is continuous’. $\endgroup$ – k.stm Nov 23 '14 at 14:44
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    $\begingroup$ Here is another suggestion: "$x+\sin(x)$ is continuous in $x$". No need to abuse the terminology. $\endgroup$ – Michael Bächtold Sep 29 '16 at 9:37
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Though I also like to use the notation "the function $x^3$" when appropriate, sometimes this causes problems. For example when I teach inverse functions, it is difficult to grasp that the $x$ in $x^3$ is not the same $x$ as in $\sqrt[3]{x}$.

I always have problems with first year students who are used to the notation that $x$ is in the domain and $y$ is in the range of the function.

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