Let $\Theta \subset \mathbb{R}^n, \mathcal{X} \subset \mathbb{R^m}$, and suppose that $C: \Theta \rightrightarrows \mathcal{X}$ is a correspondence defined by $f: \Theta \to \mathbb{R}^d$ as follows:

$$ C(\theta) = \{ x\in \mathcal{X} \mid f(\theta, x) \geq \mathbf{0} \} $$

Is it true that 1) if $f$ is continuous then $C$ also is, and 2) if $C$ is continuous then $f$ also is?

(Here continuity for correspondences is defined as upper and lower hemicontinuity.)