Map between circles

Question

Let $C_1, C_2 \subset \mathbb{R}^2$ be concentric circles in the plane. Suppose that $C_1$ bounds $C_2$. Let $f: C_1 \rightarrow C_2$ be a map such that for some $y \in C_2$, $f(x) = y$ for all $x \in C_1$ and that $x$ and $y$ must be connected by a path that is entirely contained in the interior of the region bounded by $C_1$ and that does not intersect $C_2$. Can $f$ be continuous?

EDIT:

I should probably phrase this differently. If $f$ is defined as above, can the sequence of paths under the condition above be continuous?

Answer 1

Every constant map is continuous, so in particular your map is continuous.

However, you cannot choose such paths from each $x\in C_1$ to $y$ such that they give a continuous map $F:C_1\times [0,1]\to B$, where $B$ is the closed annulus between $C_1$ and $C_2$. To see this, let $p:B\to C_1$ be the radial projection and consider the composition $pF:C_1\times [0,1]$. Then $pF(x,0)=x$ for all $x$ and $pF(x,1)=p(y)$ for all $x$, so $p$ is a homotopy from the identity map $C_1\to C_1$ to a constant map. Since $C_1$ is not contractible, no such homotopy exists.

Answer 2

Yes, for instance the radial map $$ x^2 + y^2 = h\, y $$ where $ h <1 $ maps all circles on x-axis to its dilated counterparts.

EDIT1:

Or if non-intersection among circles is very strict, the $\tau $ circles in bipolar co-ordinate map or grid (Fig 181):

$$ x^2- 2 x h + y^2 +T^2 =0 $$

The circles are distinct, non touching, same constant tangent length $L$ , variable set constant $h$ for each circle... but this example is from non-euclidean hyperbolic geometry, out of the given tag.

Bipolar_Tau_Circles