[jason1990]

Consider the vector v = (3,1,7,9). Normally, we use subscript notation to denote its components:

v_1 = 3
v_2 = 1
v_3 = 7
v_4 = 9

But sometimes we use parentheses instead:

v(1) = 3
v(2) = 1
v(3) = 7
v(4) = 9

When we use parentheses, it makes it clear that the vector v is really just a function whose domain is {1,2,3,4}. Conversely, any real-valued function whose domain is {1,2,3,4} is really just a vector in R^4.

Now consider the infinite sequence a = {2,4,8,16,32,...}. This can be regarded as a vector with a countably infinite number of components. Normally, we use subscript notation for the elements of the sequence:

a_1 = 2
a_2 = 4
...
a_n = 2^n
...

But sometimes we use parentheses instead:

a(1) = 2
a(2) = 4
...
a(n) = 2^n
...

When we use parentheses, it makes it clear that the sequence a is really just a function whose domain is {1,2,3,...}. Conversely, any function whose domain is {1,2,3,...} is really just an infinite sequence. In other words, an infinite sequence can be regarded either as a vector with countably many components (one component for every natural number) or as a function whose domain is {1,2,3,...}. Mathematically, they are the same thing. There is no mathematical difference between functions and vectors. A function is just a vector that has one component for every point in its domain.

Extending this reasoning, any function whose domain is R is really just a vector that has one component for each real number. That is, it is a vector that has uncountably many components.

A function of 4 variables is a function whose domain consists of vectors in R^4. That is, a function of 4 variables is a function of functions on {1,2,3,4}. A function of countably many variables is a function whose domain consists of sequences. That is, a function of countably many variables is a function of functions on {1,2,3,...}. Likewise, an example of a function of uncountably many variables would be a function of functions on R. rufus has given several examples.