Help with Notation: Understanding x(.) & C(.)

[emergentecon]

Having trouble understanding some notation:

x(.) or C(.) (the dot is in fact in the center i.e mid-way vertically in the brackets).

I have never run into this before? What does it mean if x is a function of "."?

 

[haruspex]

Having trouble understanding some notation:

x(.) or C(.) (the dot is in fact in the center i.e mid-way vertically in the brackets).

I have never run into this before? What does it mean if x is a function of "."?

More context please.

 

[emergentecon]

More context please.

Apologies, I thought it was something along the lines of f' being the first derivative of the function, and nothing else (to my knowledge).

And I quote "C(.) is a choice rule (technically a correspondence) that assigns a nonempty set of chosen elements C(β) ⊂ β for every budget set β ∈ B."

 

[haruspex]

Apologies, I thought it was something along the lines of f' being the first derivative of the function, and nothing else (to my knowledge).

And I quote "C(.) is a choice rule (technically a correspondence) that assigns a nonempty set of chosen elements C(β) ⊂ β for every budget set β ∈ B."

Seems like it is just a way of saying "C is a function".

 

[emergentecon]

Seems like it is just a way of saying "C is a function".

Ok, thanks!

 

[emergentecon]

Seems like it is just a way of saying "C is a function".

Someone has mentioned to me that it indicates that a function can only take a single argument . . . without specifying the specific argument?
Not sure if this is true.

 

[emergentecon]

Having trouble understanding some notation:

x(.) or C(.) (the dot is in fact in the center i.e mid-way vertically in the brackets).

I have never run into this before? What does it mean if x is a function of "."?

I got a formal answer:

f(.) means that we have a univariate (one variable) function.
The difference between f(x) and f(.) is that that, f(x) denotes a univariate function depending from the specific one-variable x, whilst f(.) denotes an one variable function depending from any one-variable we like.
We can write, for instance, f(x) or f(y) or f(z) etc

 

[Ray Vickson]

I got a formal answer:

f(.) means that we have a univariate (one variable) function.
The difference between f(x) and f(.) is that that, f(x) denotes a univariate function depending from the specific one-variable x, whilst f(.) denotes an one variable function depending from any one-variable we like.
We can write, for instance, f(x) or f(y) or f(z) etc

Be very careful about what you regard as a "variable". I have seen functions C(.) whose arguments are n-vectors, so we really have $C(x_1,x_2, \ldots x_n)$, but with the n arguments bundled together into a single "vector" $\vec{x} = (x_1,x_2 \ldots,x_n)$. In that sense, C is a function of the single "variable} $\vec{x}$. I have also seen functions C(.) whose arguments are functions themselves (such things are usually called functionals), so in a sense are functions of infinitely many variables. But, again, these several variables are all bundled together into a single object $x(.)$, and that is plugged into the formula for C.

 

[Fredrik]

The dot notation is often used when you want to define a function from one that's already defined, without coming up with a new function symbol. For example, if $f:\mathbb R^2\to\mathbb R$ and $y\in\mathbb R$, then $f(\cdot,y)$ denotes the function $x\mapsto f(x,y)$ with domain $\mathbb R$, i.e. the function $g:\mathbb R\to\mathbb R$ such that $g(x)=f(x,y)$ for all $x\in\mathbb R$. So for all $x\in\mathbb R$, we have $f(\cdot,y)(x)=f(x,y)$. The dot is just telling you where to put the "input".