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The standard way to write $ \text{ y is a function of x} $ is

$ y = f(x) $

This is taken to mean that $y$ is the value of function $f$ evaluated at $x$. For simplicity let's take $f$ to be some mapping, $ f:\Bbb R\to\Bbb R$.

I cannot understand if mathematics authors are justified in using the notation

$y = y(x)$

to declare that $y$ is a function of $x$. The reason is a type mismatch, it cannot be be possible for $y$ to be a binary relation, as well as some element in the codomain of the binary relation.

Is the notation above commonly accepted? I have seen it in a few published papers, and am not sure whether it is an abuse or has some sound mathematical reasoning.

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It is abusive and only analysts and physicists write such things. –  Git Gud Mar 12 '13 at 23:07
It is a slight abuse. On the other hand, all kinds of abuses are tolerated, so why not this one. With respect to type mismatches, the integer 1 plus the real number $\pi$. You see $1+\pi$ is not necessarily defined if 1 is an integer, but this is a shorthand, even if it is a type mismatch. My point is that type mismatches are not necessarily mortal sins. –  Baby Dragon Mar 13 '13 at 15:17
Mathematicians write for other human beings who can understand what is going on, so that abuse of notation is not really a big deal in practice. –  KCd Jun 19 '14 at 1:16

3 Answers 3

There is nothing odd or abusive in this notation. The heart of the problems seems to lay in the fact that the notation $$ = $$ is understood in mathematics in two different senses.

First, it is the equality operator, meaning that expression of the form $$ A=B $$ is true if $A$ is equal to $B$.

Second, it is an assignment operator, meaning that the expression of the form $$ A=B $$ amounts to "$A$ is defined to be the same as $B$, which we already know". It is this second meaning is usually supposed in the expressions of the form $y=y(x)$. For instance, I often write in my papers something along the line: "Consider a system of ODE $$ \dot x=f(x), $$where $x=x(t)$" emphasizing that letter $x$ in my text denotes function $x(t)$.

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Excellent take on this issue. +1 –  James S. Cook Jun 19 '14 at 1:49
There is still something of an abuse: in your last sentence, you write "letter $x$ in my text denotes function $x(t)$", but $x(t)$ is not a function but the value of the function $x$ at $t$... –  Pece Jun 19 '14 at 13:56

Without knowing the context to which you are referring I would say the author doesn't wish to a lot of $(x)$'s. If, for example, there is integration or differentiation with respect to $x$ then you have to take the fact that $y$ is a function of $x$ into account. On the other hand if there is integration or differentiation with respect to $t$ then perhaps you can consider $y$ to be a constant. This answer is speculative because we are not given a complete example.

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This bothers me too and I never do it in my private life. However, when teaching calculus it's unavoidable. To keep my inner-stickler happy, I secretly think to myself that $x=\DeclareMathOperator{id}{id}\id_X$ where $X$ is the domain of $y$. Thus we have $$ y=y\circ\id_X=y\circ x $$

Defining $y(x)$ as $y\circ x$ then justifies the notation $y=y(x)$ and I don't have to think about elements.

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This is an intersting idea; how does $y = y \circ id_{X}$ solve the problem? Are you saying that $y$ is a function on the both sides, and that $y(x) = (y \circ id_X) (x)$ for all $x$ (i.e is this a statement of equivalence, rather than assignment)? –  jesterII Apr 15 at 17:20
@jesterII No I'm not saying that. –  Brian Fitzpatrick Apr 16 at 3:52

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