What is the difference between variable, argument and parameter?

Question

I'm sure that these terms should be different since there exists a difference between parameter and argument in computer science but I'm not sure about their differences in math.

Answer 1

Consider family of functions $\{f_a : a\in [0,1]\}$, where $f_a:\mathbb{R}\rightarrow \mathbb{R}$ is given by $$f_a(x)=x^2 - a.$$ When you take some $x\in \mathbb{R}$, this is a variable. This variable becomes an argument for function $f_1$ if you begin to consider $f_1(x)$. In this expression, $1$ is a parameter.

More generally, parameter is a "selector" from a family of "similar" functions. Variable is basically any element from any set. If you take variable from domain of some function to consider value of this function at given point, then variable becomes an argument for the function.

I hope it made this topic more understandable.

Remember, that these terms, though diffeerent, are sometimes used interchangeably. For example, you can consider a function of two variables $f(x,y)$ as a function of one variable $y$ with parameter $x$.