# Notation for a function definition

I came across the following notation in a research paper

Suppose we have a function $$f(x) \in [1,l] \rightarrow \mathbb{R}$$

It is the first time, I am looking such a notation. And the paper is using the same notation for all functions used in it.

Is it same as $$f(x) : [1,l] \rightarrow \mathbb{R}$$

If yes, is it an abuse of notation?

If no, what does the notation mean?

• What a weird notation. Commented Feb 15, 2021 at 9:08
• It would probably be helpful if you gave some idea of when the paper was published (decade identification is probably enough -- 1950s, 1960s, 1970s, 1980s, etc.) and gave the name of the journal. For instance, I would imagine this would be more peculiar if published in Transactions of the AMS in the 1990s than in some mostly unknown journal less than 20 years old. Commented Feb 15, 2021 at 11:10
• @DaveL.Renfro It is a recent paper, around 2016... Commented Feb 15, 2021 at 11:25
• It must be stressed that $f(x)$ is not a function. $f$ is. Commented Feb 23, 2021 at 3:25

In the paper you're using, I understand it means that $$f$$ belongs to the set of all functions defined from the set $$[1,l]$$ to the set $$\mathbb{R}$$, to be said, $$[1,l]\to\mathbb{R} := \{f \text{ function }: f \text{ is defined from [1,l] to \mathbb{R}}\}$$ so it works the same way as defining $$f$$ the usual way, to be said: $$f\in[1,l]\to\mathbb{R} \equiv f:[1,l]\to\mathbb{R}$$