04 - Path Connectedness of Images

Path Connectedness of Images

Let $U\subseteq {\mathbb{R}}^2$ be path-connected and $f:U\to {\mathbb{R}}^2$ be continuous then

$$f(U)\text{ is path-connected}$$

Proof

Let $v_0,v_1\in f(U)$ then

$$v_0=f(u_0)\text{ and }{v}_1=f(u_1)\text{ for some }{u}_0,u_1\in U$$

As $U$ is path connected then there is a path $\gamma$ joining $u_0$ and $v_0$

Then $f\circ \gamma$ gives the required path between $v_0$ and $v_1$

To show continuity use this to show the composition of two continuous maps is also continuous