How to curry and uncurry a function.

How to curry and uncurry a function $\mathbb{R}\rightarrow\mathbb{R}^N$. 

The functions $A\rightarrow{B}^C$ corresponds to functions of the form $A\times{C}\rightarrow{B}$. Going from the function $A\times{C}\rightarrow{B}$ to the function $A\rightarrow{B}^C$ is called currying, while going from the function $A\rightarrow{B}^C$ to the function $A\times{C}\rightarrow{B}$ is called uncurrying.

Suppose $f$ is a function $A\times{C}\rightarrow{B}$ of two arguments, then we say its curried form $f'$ (not derivative) is a function $A\rightarrow{B}^C$ of one argument but whose values are themselves functions. For $a\in{A}$, $f'(a)$ is a function $B\rightarrow{C}$ whose value at $b\in{B}$ is $f'(a)(b)=f(a,b)$.

So a function $\mathbb{R}\rightarrow\mathbb{R}^N$ corresponds to a function $\mathbb{R}\times\mathbb{N}\rightarrow\mathbb{R}$. Take any function from $\mathbb{R}\times\mathbb{N}$ to $\mathbb{R}$, say $f(x,n)=x+n$, then it is curried from $f'$ whose values at $x$ denoted by $f'(x)$ is the function $\mathbb{N}\rightarrow\mathbb{R}$ whose value at $n$ is $x+n$. Symbolically, $f'(x)(n)=f(x,n)=x+n$.