Definition of a curve (Jordan Curve Theorem)

Question

In my graph theory course, we are talking about planar graphs, and we studied the definition of a curve prior to talking about the Jordan Curve Theorem. However, I'm having a bit of trouble understanding the definition provided to us. Here it is

I sort of understand what it's trying to say. A curve is similar to a line segment. A closed curve is similar to a circle. Though, I'm not sure what "continuous image" means.

Also, is the Jordan Curve Theorem just saying that a closed loop partitions the plane into an "inside" and "outside"?

Sorry for the trivial question.

Thanks

Answer 1

You got it correctly -- the JCT says that a closed non-self-intersecting curve partitions the rest of the plane into inside and outside.

Continuous image means that there is a continuous function $f:C(0,1) \to \mathbb{R}^2$, where your closed curve is the image of $f$ (where $C(0,1)$ denotes the unit circle).