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There are varies questions on this site concerning with the meaning of $\cdot$ in the expression $f(\cdot)$, for example this one, this one and this one. However, I'm not sure when using this notation is acceptable. Is the $\cdot$ notation correct in each of the following cases:

(a) Using $\cdot+1$ to replace the function $x\mapsto x+1$?

(b) Using $3\cdot$ to replace the function $x\mapsto 3x$ (I must admit that this looks highly unacceptable)?

(c) The function $\displaystyle\sum_{1\le i<j\le\cdot}(j-i)$ defined over $\mathbb{N}^*$?

(d) The function $\begin{cases}\cdot^2,&\cdot>1\\ 5,&\cdot\le 1\end{cases}$?

Any help appreciated.

Jianing Song
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1 Answers1

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It's used to represent a free variable when the subject of discussion is the operator without reference to operand.

Generally it's only used inside delimiters of some kind for clarity's sake, but technically it's okay to use it wherever a pronumeral might be used although not advisable.

Localth
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