Doubt in Extension theorem of Sobolev space on Half space

Question

I was reading the Proof of Extension theorem of Sobolev space in that Author constructed the following

He takes $\phi \in C_c^{\infty} (R^n )$ and define $\psi(x',x_n)=\phi(x',x_n)+\phi(x',-x_n)$ for fixed $x_n>0$

Isn't compactly supportedness of $\phi $ implies that of $\psi $ in the upper half-space? But Author it is not true. I do not find any argument to show a contradiction

Any Help will be appreciated

Answer 1

This $\psi$ will not be continuously differentiable in general, as the support of $\phi$ might cross the hyperplane $\{ x_n=0\}$.