Let $E: C_b^1(\mathbb R_+ \times \mathbb R) \rightarrow C_b^1(\mathbb R_+ \times \mathbb R) $
where
$$(Ef)(x_1,x_2):= \begin{cases} f(x_1,x_2) & \text{ if }x_1\ge0 \\ \alpha f(-x_1,x_2) + \beta f(-0.5x_1,x_2) & \text{ if }x_1 \lt 0\end{cases}$$
Give values for $\alpha$ and $\beta $ and prove that $E( C_b^2(\mathbb R_+ \times \mathbb R)) $ is not a subset of $C_b^2(\mathbb R_+ \times \mathbb R)$
What I got so far:
$E(f)$ has to be continuous in $0$ which leads me to the following equation:
$$\alpha +\beta =1$$
$E(\frac{d}{d_{x_1}}f)$ has to be continuous in 0 which leads to the equation: $$-\alpha+-\frac{1}{2}\beta =1$$
Combining the two I get: $\alpha =-3 \quad \beta=4$
Continuing the same "thing" for $f \in C^2$
$$\alpha+\frac{1}{4}\beta=1=-2$$
which is a contradiction.