(a) Suppose that $g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ is a continuous function such that there exists a $K>0$ with $\|g(\mathbf{x})-\mathbf{x}\| \leqslant K$ for all $\mathbf{x} \in \mathbb{R}^{2}$. By constructing a suitable map $f$ from the closed unit disc into itself, show that there exists a $\mathbf{t} \in \mathbb{R}^{2}$ with $g(\mathbf{t})=\mathbf{0}$.

(b) Show that $g$ is surjective.

(c) Show that the result of part (b) may be false if we drop the condition that $g$ is continuous.