Let $S^1 = \{(x, y) \in \Bbb R^2 : x^2 + y^2 = 1 \}.$ Let $D = \{(x, y) \in\Bbb R^2 : x^2 + y^2 \le 1 \}$ and $E = \{(x, y) \in\Bbb R^2 : 2x^2 + 3y^2 \le 1\}$ be also considered as subspaces of $\Bbb R^2.$ Which of the following statements are true?
a. If $f : D \to S^1$ is a continuous mapping, then there exists $x \in S^1$ such that $f(x) = x$.
b. If $f : S^1 \to S^1$ is a continuous mapping, then there exists $x \in S^1$ such that $f(x) = x$.
c. If $f : E \to E$ is a continuous mapping, then there exists $x \in E$ such that $f(x) = x$.