Why is this not a linear space of functions?

Question

Consider

Why is $D$ not a linear space of functions? It is said that because of the non-zero endpoints, $D$ is not a linear space, because addition is not closed. But I think I'm missing something about this concept. How does one show that $D$ is not closed under addition?

Answer 1

Because if $\theta_B\neq 0, (\theta,\phi), \in D, 2\theta(0)=2\theta_B\neq \theta_B$ so $2(\theta,\phi)$ is not in $D$.