Jordan curve theorem: Can every point on the curve be reached from each region? The Jordan-Schönflies theorem states that any simple closed curve $C$ divides the plane into two connected regions $U$ and $V$, and that any point $x \in C$ is a boundary point of both $U$ and $V$.  A stronger statement would be that for any point $x$ in $C$, the set $U \cup V \cup \{x\}$ is pathwise connected.  Equivalently, any point in $U$ can be connected to $x$ by a continuous path lying entirely inside $U$ (except for its ending point, of course) and likewise for $V$.
I haven't run across any proof of this stronger statement.  Has it been proved?  If not, is there a counterexample?
 A: The answers by Kent and Eremenko are, of course, absolutely correct. Here is the more general statement: Let $M$ be a closed connected topological $n-1$-dimensional manifold embedded in ${\mathbb R}^n$ (note that $M$ can be wild for $n\ge 3$). Let $U$ be a component of ${\mathbb R}^n -M$. Then for every point $x\in M$ there exists a continuous path $p: [0,1)\to U$, so that
$$
\lim_{t\to 1} p(t)=x$$
In particular, $({\mathbb R}^n -M) \cup \{x\}$ is path-connected. The proof is based on the following observation:
Observation. There exists a function $\phi(r), r\in [0, \infty)$, satisfying $\phi(r)\ge r$ and
$$
\lim_{r\to 0+} \phi(r)=0$$
so that: For every $r>0$ the map 
$$
\tilde{H}_0(U \cap B(x, r))\to \tilde{H}_0(U \cap B(x, \phi(r)))
$$
(induced by inclusion) is zero. In other words, any two points in $U \cap B(x, r)$ can be connected by a path in $U \cap B(x, \phi(r))$. 
The above observation follows from the Poincare/Alexander duality in ${\mathbb R}^n$ 
(in conjunction with the fact that $M$ is a manifold).
Given the above observation, take a sequence $x_k\in U\cap B(x, \frac{1}{k})$ and 
construct the path $p$ by concatenating paths $p_i$ in 
$$
B(x, \phi(\frac{1}{i}))\cap U
$$
connecting points $x_i, x_{i+1}$, $i\in {\mathbb N}$. 
A: Yes. One way to prove this (and the Jordan theorem too) is to use Complex Variables:-) 
A good reference is Milnor,
MR2193309 Dynamics in one complex variable. Third edition. Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006. 
A chapter in this book contans the best exposition of these questions that I know.
A: Let y be a point in, say, the exterior E of C, and x be a point in C. Then x belongs to the closure of E, hence, there is a sequence of points {x_n} in E which converges to x. As E is connected, there is a chain 
y --- x_0 --- x_1 --- x_2 ---
of continuous curves lying entirely within E. 
To see that their union connects y to x, we have to use a trick. 
Namely, after a line s_0 connecting y to x_0 within E is defined, let r_0 be its distance to C, S_0 be the open (r_0/2) nbhd of s_0, and X_0 be the open (r_0/2) nbhd of x_0 - now we claim that the line s_1 connecting x_0 to x_1 be defined so that it does not contain points in S_0 except of those in X_0 - which is easily possible by path-connectedness. 
A suitable iteration of this trick yields the result. 
