1. ## continuous curve?

I'm taking a history of mathematics course and I have a question on my assigment: Give the definition, "the curve g:[0,1] to R^2 (= R x R) is continuous" (the elements of R x R are ordered pairs (x, y) of reals; each such pair represents, via a chosen Cartesian coordinate system, a point in the plane). I know the definition of a continuous function f using the epsilon-delta notation (and much simpler definitions), but how can I define this, a curve with R^2? Is it similar?

This is also pretty random, but does anyone know if Euclid had a theorem similar to the Intermediate Value Theorem and how he proved it?

2. In general given a vector-valued function $\displaystyle \bold{f}: \mathbb{R}^n \mapsto \mathbb{R}^m$ we say $\displaystyle \bold{f}$ is continous at (vector not number here) $\displaystyle \bold{a}\in \mathbb{R}^n$ iff $\displaystyle |f(\bold{x})-f(\bold{a})|<\epsilon \mbox{ for }|\bold{x}-\bold{a}|<\delta$ where $\displaystyle | ~ |$ is the 'Euclidean norm' defined for in the following way: "if $\displaystyle \bold{a} = (a_1,...,a_n)$ then $\displaystyle |\bold{a}| = \sqrt{a_1^2+...+a_n^2}$". Does that make sense?

I wish I was taking History of Math couse.

3. We can think of your problem like I stated above or in a different way.

We have $\displaystyle \bold{f}:\mathbb{R}\mapsto \mathbb{R}^2$ so we can view $\displaystyle \bold{f} = (f_1,f_2)$, i.e. the components of the vector-valued function. And each $\displaystyle f_1,f_2$ is continous. Maybe this is an easier way to think of this.