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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
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1 Answer

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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