Reciprocal spiral with its endpoint ($\{t^{-1}{e}^{2\pi it}:t\ge 1\}\cup \{0\}$)

The map $F(s)=s{e}^{2\pi i/s}$ for $s>0$, $F(0)=0$, is continuous on $[0,1]$, since $|F(s)|=s$. Its image is the displayed spiral together with its endpoint. The image is compact and path-connected: between images of $s_0,s_1$ the explicit path is $F((1-u)s_0+u{s}_1)$.