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Intuitively, the parametric equation $z = (x(t), y(t))$ seems to only be able to generate one-dimensional objects, i.e. curves.

Let $x(t)$ be "the odd-indexed digits of the real number $t$", and let $y(t)$ be "the even-indexed digits of the real number $t$". So for example, $x(123.4567) = 13.57$ and $y(123.4567) = 2.46$.

Now let's say we're given the parametric equation $z = (x(t),\ y(t))\ \ \ \forall\ t \in \mathbb{R}$.
This equation, essentially, represents a bijective mapping between $\mathbb{R}$ and $\mathbb{R} \times \mathbb{R}$.

My questions are:

  1. Can $z$ be called a (two-dimensional) "surface" (instead of a curve)? (Why?)

  2. If so, is it correct to say the surface is continuous? (Why?)

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+1 Nice question, Mehrdad. – Babak S. Jan 18 '13 at 4:24
@BabakSorouh: Thanks! – Mehrdad Jan 18 '13 at 4:36
up vote 5 down vote accepted

You have to make sure that it is well-defined, which means you need to decide which representation to use for $t=0.499999....=0.5000...$. You'd also have to prove the function is continuous. The tricky point is at the same values of $t$ where it wasn't well-defined.

Even then, the function is not $1-1$. If you take $t_0=0.409090909090...$ and $t_1=0.500000....$ they get sent to the same value. What you have here is called a "space-filling curve."

That said, there is no meaning to "continuous surface" in this sense. A "curve" is not just the image of the map $(x(t),y(t))$, but also the path through that image. Under that view, this is still a continuous curve, not a continuous surface.

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