Curves in $\mathbb R^n$ and boundary points Thinking of a curve as a 1-dimensional continuum, intuitively one would think that a curve can never have this property. Perhaps intuition is correct and this question is somewhat naive, but here it is anyway:

Can we find a curve or trajectory $A \subset \mathbb R^n$ such that there is at least one point on the curve that is not a boundary point (under the Euclidean topology)? Infinitely many such points? The entirety of the curve?

I feel that such a curve would probably have a fractal construction in some sense, but I can't fathom what it would look like.
 A: Why not? Check out space-filling curves. Such a beast is a continuous, surjective function $[0,1]\to [0,1]^2$. Since the function is surjective, no point in the preimage of the interior of the unit square will map to a boundary point of the image in $\mathbb{R}^2$.
Here's a gif animating the first several iterations of Hilbert's construction of such a curve.

The reason your intuition is wrong is that your repertoire of examples is filled with differentiable functions, not arbitrary continuous functions. In fact, differentiability is an incredible constraint on how a curve can behave. If your function is differentiable at a point, that means as you zoom in, there's only one direction, one speed it can be going: The direction and speed of the derivative. 
But there are so many other directions it could go at that point if it were merely continuous, not differentiable. 
An arbitrary continuous curve is not differentiable; it displays "fractal" behavior, in the sense that no matter how far you zoom, it will be jagged. This can be made precise in a couple of different ways, but the gist is that almost every continuous function is nowhere differentiable.
