Continuous image of a disk — can't be two disjoint pieces

Let $A=\{(x,y)/y^2\le x^2+y^2\le1\}$ (the condition $y^2\le x^2+y^2$ holds automatically, since it just says $x^2\ge0$, so A is simply the closed unit disk) and $f:A\to\mathbf R$ be a continuous map. Which of the following is false?