Is parametric form of a given function unique?

Question

1. Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?)

2. Given any function, is parametric form uniquely determined?

Answer 1

1. No. Any parametric curve is a connected set (since it is the image of a connected set under a continuous mapping). But the set $\{(x,y) \in \mathbb{R}^2 : x^2 = 1\}$ consists of two disconnected lines, so it can't be represented using parametric equations (using continuous functions, anyway).

2. No. The parameterizations $t \mapsto(0,t)$ and $t \mapsto(0,1-t)$ both represent the same curve, for example. Even if you restrict $t$ to $[0,1]$, it's still true that both parameterizations give you the same curve segment.

See also this question, which provides an example of two different parameterizations of a quadrant of a circle.

Answer 2

1. In general, no. For example, many piecewise functions have no parametric forums. You will have to be more specific about this and if you clarify, I will edit this answer. For example, consider the Dirichlet function. I don't know if you consider that in the set of functions you are asking about.

2. No. Quick counterexample. Take $ y = x $. The parametric equations $$ \begin {eqnarray*} x &=& t, \\ y &=& t \end {eqnarray*} $$and $$ \begin {eqnarray*} x &=& 2t, \\ y &=& 2t \end {eqnarray*} $$There are many other counterexamples.