Let $c>0$ and let $X$ be a **subspace** of $C^{\infty}_c((0,1);\mathbb R)$ such that for any $\phi \in X$ there holds:
$$ \int_{r}^1 \phi'(t)^2\,dt \leq c\,\int_{r-r^2}^1 \phi(t)^2\,dt \quad \forall\, r\in (0,1).$$
Is it true that $X$ must be finite dimensional?