# Is there any way to refer to a function $f$, without defining $f$? [closed]

If I have to refer to a function $$f$$ without defining it explicitly, what should I write? Or is there no notation I can use but explicitly defining $$f$$?

• I'm not sure why you have a problem here - it is common in advanced math to write about functions you never explicitly defined. It comes with the territory. Aug 12, 2021 at 20:02
• dear my2cents, do you know what the domain and codomain of the function are going to be? if so, (suppose they are $X$ and $Y$, respectively) then you can just say "let $f:X\to Y$ be a function", or "let $f$ be a function from $X$ to $Y$" Aug 12, 2021 at 20:04
• ahhh, I see; I personally think that it is best for clarity to write something like "for each $i\leqslant n$, define $f_i:\mathbb{R}\to\mathbb{R}$ by $f_i(x)=e^{\lambda_i x}$." but ultimately it's up to you; what you've written in that sentence seems quite clear already to me, although maybe others would disagree Aug 12, 2021 at 20:22
• And then after, if I need to refer to $f_i$, I can just write $f_i$ (e.g. $Tf_i$, which is in my next sentence). Aug 12, 2021 at 20:26
• You could also use a notation like $e^{\lambda(\cdot)}$ or $x \mapsto e^{\lambda x}$ for this purpose. However I think just defining $f_i$ explicitly as @atticus says would probably be best if you want to refer to them later. You could also use a notation like $f_{\lambda}(x) = e^{\lambda x}$ if you don't want as many indices. Aug 12, 2021 at 20:40

I think what you’re looking for is the notation $$x \mapsto x^2$$ which allows you to write the function $$f$$ where $$f(x) = x^2$$ without giving it a name. (This notation also leaves out the domain and codomain which might be fine if they’re clear from context or if you mention them in the surrounding text.)
It seems like writing "Let $$f: V\to W$$ be a function defined by [formula]" or similar is clearest.