Are the partial derivatives of any order n and below of a continuously differentiable function of order n with a compact supported set bounded?

Question

Let $C_c^n\left(\Omega\right)$ be the set of all the continuously differentiable function of order n with a compact supported set included in $\Omega$. I've already known that $\forall f\in C_c^n\left(\Omega\right)$, $f$ is bounded. But, is any partial derivative of order $m$ that less than or equal to $n$ of $f$ also bounded? That is $\left|D^\alpha f\right|\le M,\left|\alpha\right|=m\le n$, where $\alpha$ is a multi-index.

Answer 1

Since $f$ has compact support, the derivatives $D^\alpha f$ have compact support for $|\alpha|\leq n$. Since the derivatives are continuous on this compact set, $D^\alpha f$ are bounded for all $|\alpha|\leq n$.