Let $$\mathcal{C}_0(\mathbb{R}) = \left\{f : \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ is continuous}, \lim_{x \rightarrow \infty} |f(x)| = 0 \text{ and } \lim_{x \rightarrow -\infty} |f(x)| = 0\right\},$$ and $\mathcal{C}_0^\infty(\mathbb{R}) = \left\{f \in \mathcal{C}_0(\mathbb{R}) \mid f \text{ is infinitely differentiable}\right\}$. Let $\phi \in \mathcal{C}_0(\mathbb{R})$. For $f \in \mathcal{C}_0(\mathbb{R})$, define $\phi^*(f) = f \circ \phi$. (A) Show that $\phi^*(f) \in \mathcal{C}_0(\mathbb{R})$ if $f \in \mathcal{C}_0(\mathbb{R})$. (B) If $\phi^*\left(\mathcal{C}_0^\infty(\mathbb{R})\right) \subseteq \mathcal{C}_0^\infty(\mathbb{R})$, then show that $\phi$ is infinitely differentiable.
Let
$$\mathcal{C}_0(\mathbb{R}) = \left\{f : \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ is continuous}, \lim_{x \rightarrow \infty} |f(x)| = 0 \text{ and } \lim_{x \rightarrow -\infty} |f(x)| = 0\right\},$$
and $\mathcal{C}_0^\infty(\mathbb{R}) = \left\{f \in \mathcal{C}_0(\mathbb{R}) \mid f \text{ is infinitely differentiable}\right\}$.\\
Let $\phi \in \mathcal{C}_0(\mathbb{R})$. For $f \in \mathcal{C}_0(\mathbb{R})$, define $\phi^*(f) = f \circ \phi$.\\
(A) Show that $\phi^*(f) \in \mathcal{C}_0(\mathbb{R})$ if $f \in \mathcal{C}_0(\mathbb{R})$.\\
(B) If $\phi^*\left(\mathcal{C}_0^\infty(\mathbb{R})\right) \subseteq \mathcal{C}_0^\infty(\mathbb{R})$, then show that $\phi$ is infinitely differentiable.