Function that returns a function Is there a notation for a function that returns another function? Maybe something like $f(x) = (y) = y + x$ i.e. the function being returned would return it's input plus $x$, e.g. $f(2)(3) = 5$?
Is the way I have written it correct?
 A: Let $\mathcal{F}$ be the set of functions $\Bbb{R}\to \Bbb{R}$. If I'm understanding you correctly, what you're talking about is a function $f:\Bbb{R}\to \mathcal{F}$, such that for all $x\in \Bbb{R}$, $f(x)\in \mathcal{F}$ is that function $\Bbb{R}\to \Bbb{R}$ such that for all $\alpha\in \Bbb{R}$, we have
\begin{align}
[f(x)](\alpha):= \alpha + x
\end{align}
A slightly quicker way of writing this is that for all $x\in \Bbb{R}$,
\begin{align}
f(x):= (\Bbb{R}\ni \alpha \mapsto \alpha + x)
\end{align}
Or yet another way of writing this is: $f:\Bbb{R}\to \mathcal{F}$,
\begin{align}
x\mapsto (\alpha \mapsto \alpha + x)
\end{align}
In words, this is saying that $f$ is that function which maps an element $x\in \Bbb{R}$ to the function $\alpha\mapsto \alpha + x$ from $\Bbb{R}\to \Bbb{R}$.

So, yes we have $[f(2)](3) = 2+3 = 5$. If you want to be slightly more economical with your bracketing then sure $f(2)(3) = 5$ is also true. But if you're ever writing this stuff down for someone else to read then make sure you write "$f:\Bbb{R}\to \mathcal{F}$" before hand to avoid any misunderstanding. Because if you don't say anything before hand then people will automatically assume that $f$ is itself a function $\Bbb{R}\to \Bbb{R}$ so when they see $f(2)(3)$ they'll either think this is weird notation/ a typo/ a multiplication of $(f(2)) \cdot (3) = 3 \cdot f(2)$, which is of course not what you intended.
