Difference between curve and a function of two variables.

Question

Please help as I could not find out what I am missing. I can understand to an extent https://en.wikipedia.org/wiki/Curve what it means to be a curve. But I cannot find what I loose if I try to define curve as function of two variable. For simplicity consider a curve in X-Y plane in such case can't we think of curve as a function of two variable?

Answer 1

You'll end up with a surface in $\mathbb R^3$ if you try to define a curve $z = f(x,y)$ since $(x,y)\to f(x,y)$

Take the parabola as in the article for example

We have no problem looking at $y = x^2$ in $\mathbb R^2$, but in $\mathbb R^3$, the equation is independent of $z$, so $z$ takes on all values, which extends the parabola indefinitely in both $z$ directions.

Making the parabola just a curve and not a surface in $\mathbb R^3$ requires parameterized equations like

$x(t)=t$

$y(t)=t^2$

$z(t)=0$