Let us assume that I have a function $f_L$ defined on a grid of mesh size $h_L$ such that $f_L(x)= \sum_{i=1}^{n_{L}} f_{i,L} \phi_{i,L}(x)$, where $\phi_L$ are piecewise polynomials of order $q$. I have a problem where I need, for some analysis, to assume that $$\mathrel{\Big|} \mathrel{\Big|} f_{L}^{(p+1)} \mathrel{\Big|} \mathrel{\Big|}_{L^{2}(I)} \leq C$$
where $f_{L}^{(p+1)} $ is the derivative of order $p+1$ of $f_L$, $q \geq p+1$ and $C$ is a constant independent of $h_L$.
Is it reasonable to have such assumption or I need to guarantee additional requirements to fulfill that assumption. Thanks.