The question was to show that $supp (D^{\alpha}f) \subset supp (f)$ if $f\in C_c^k(\Bbb R^n)$
My teacher's solution was, Let $x \in (supp f)^c \implies f(x)=0 \implies D^{\alpha}f(x)=0$ (since $(supp f)^c$ is open)
I didn't understand this last step, I mean even if $(supp f)^c$ is open, why does this fact give that $D^{\alpha}f(x)=0$ ??
N.B.: $supp f$ is the support of the function $f$
Maybe my question in other words is, if $f(a)=0$, what condition do we need so that we can deduce that $f'(a)=0$ ?
The topological support is closed by definition.
As the compliment of the support is open, there must exist an open neighborhood $U\ni x$ such that $f(y)=0$ for all $y\in U$. Thus all partial derivatives of $f$ must be $0$ at $x$, else the very definition of the derivative as the instantaneous rate of change would imply that $f$ must be non-zero somewhere in $U$. Think of the partial derivatives as forming the approximating plane for differentiable functions.