Consider the following sets :
$$S=\left\{(x,y)\in \mathbb R^2:x^2+y^2=1\right\}.$$
$$D=\left\{(x,y)\in \mathbb R^2:x^2+y^2\le 1\right\}.$$
$$E=\left\{(x,y)\in \mathbb R^2:2x^2+3y^2\le 1\right\}.$$
Which of the following are correct ?
(a) If $f:D\to S$ is continuous then $f$ has a fixed point.
(b) $f:S\to S$ is continuous then $f$ has a fixed point.
(c) If $f:E\to E$ is continuous then $f$ has a fixed point.
We know from Brouwer's fixed point theorem , " A continuous mapping of a closed & convex set into itself necessarily has a fixed point. "
From this theorem we have , option (c) is correct & option (b) is wrong.
But what about options (a) ?
Are there any theorem for a continuous function from one set to another set have to a fixed point?
a) note that $S\subset D$ and $D$ is closed and convex. So $f(D)\subset S\subset D$ and you can apply the Theorem.
b) depends on the function, e.g. $f(x)=x$ or $f$ is a rotation around the origin.
c) you are right.