I am reading a textbook of distribution theory.
Here is the definition of test functions: The class of test functions $\mathcal D(\Omega)$ consists of all functions $\varphi(x)$ defined in $\Omega$, vanishing outside a bounded subset of $\Omega$ that stays away from the boundary of $\Omega$, and such that all partial derivatives of all orders of $\varphi$ are continuous.
Since the vanishing requirement, we know there is no analytic function can be in $\mathcal D$ except $\varphi\equiv0$.
Then, the author said, Thus any formula for $\varphi$ must be given "in piece".
Why can he conclude this sentence? Can someone help me with understanding this?
Thanks in advance.