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?