6 (15) Let \( M \) be a compact metric space and \( \Phi : M \to M \) be such that \( d(\Phi(x), \Phi(y)) < d(x, y) \) for all \( x, y \in M \), \( x \neq y \). Show that \( \Phi \) has a unique fixed point.
7 Define the function \(f:\mathbb{R}^2\to \mathbb{R}\) by

$$f(x, y) = \begin{cases} \frac{x\sqrt{x^2+y^2}}{|y|} & \text{if } y \neq 0 \\ 0 & \text{if } y = 0 \end{cases}$$

a. (10) Prove that the function \( f \) is not continuous at the point \( (0,0) \).