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Is it wrong to represent a dependent variable and a function using the same symbol? For example, can we write the parametric equations of a curve in xy-plane as $x=x(t)$ , $y=y(t)$ where $t$ is the parameter? For me function is different from dependent variable. But, in many calculus texts, sometimes, a dependent variable and a function are represented using the same symbol, why?

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Technically, they are different, but writing this way is an acceptable "abuse of notation" because it saves on notational clutter. – Prahlad Vaidyanathan Sep 15 '13 at 20:10
Abuse of notation. Sometimes that's a good thing, sometimes bad. Different people have different opinions on when it's good and when bad. – Daniel Fischer Sep 15 '13 at 20:10
If it's wrong then I don't wanna be right. – James S. Cook Sep 15 '13 at 20:13
While $y=f(x)$ is a notation that conveys the verbal statement "$y$ is a $f$(unction) of $x$", sometimes the notation $y=y(x)$ is used in order to stress the fact that $y$ is a function of $x$. – Alecos Papadopoulos Sep 15 '13 at 20:55

When we write $y=f(x)$, we give names to the function, its input, and its output.

But not all of these names are needed at all times.

  • "let $f$ be a continuous function" names the function, but neither its input nor output.
  • "$\langle\cdot ,\cdot \rangle$ is jointly continuous" is another version of the above.
  • "$x\mapsto \sqrt{x}$ is an increasing function" names the input, but neither the output nor the function. A less correct, but common version of this statement omits the "$x\mapsto $" part.

What if we want to give names to the input and output, but not to the function itself? Writing $y=\cdot (x)$ is typographically awkward. Writing $y=y(x)$ conveys as much information, and is less likely to be mistaken for a typo. I think "$x\mapsto y$" would be a better choice, but this is less common.

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