Preliminary remark: If you don't know an answer in this general setting, I'm also interested in an answer for the case $E_1=\mathbb R^d$, $d\in\mathbb N$, and $E_2=\mathbb R$.
If $E_1$ is a topological space and $E_2$ is a $\mathbb R$-vector space, remember that $$\operatorname{supp}f:=\overline{\left\{f\ne0\right\}}\;\;\;\text{for }f:E_1\to E_2.$$
Assume $E_1$ is a $\mathbb R$-Banach space, $\Omega\subseteq E_1$ is open and $E_2$ is a normed $\mathbb R$-vector space. Please consider the following trivial result:
Lemma 1: If $f:\Omega\to E_2$ is differentiable at $x\in\Omega$ in direction $h\in E_1$ and $\{f=0\}$ is a neighborhood of $x$, then $${\rm D}_hf(x)=\lim_{t\to0}\frac{f(x+th)-f(x)}t=0\tag1.$$
Question
Are we able to conclude the following corollary?
Corollary 2: If $f:\Omega\to E_2$ is differentiable in direction $h\in E_2$, then $$\operatorname{supp}{\rm D}_hf\subseteq\operatorname{supp}f\tag2.$$
From Lemma 1, we immediately obtain $$\{f=0\}^\circ\subseteq\{{\rm D}_hf=0\}\tag3.$$ It's tempting to conclude $$\{{\rm D}_hf\ne0\}\subseteq\Omega\setminus\{f=0\}^\circ=\overline{\{f\ne0\}}=\operatorname{supp}f\tag4$$ and argue that the closure of the leftmost side must be contained in the rightmost side as well, since the latter is closed.
However, as thinking about this I realized that I'm not sure whether I'm considering the correct definition of the support. To be precise, in the definition of the support of $f$, do we need to take the closure with respect to the topology on $E_1$ or with respect to the subspace topology on $\Omega$?
Note that this question is crucial, since these closures do not coincide.
Regarding to my definition of the support, we would need to take the closure with respect to the subspace topology on $\Omega$ (and every closure which is taken in this question has to be understood with respect to this topology), but I wonder whether this is what's usually done.
Remark: Feel free to add any kind of continuity assumption, if necessary to obtain the desired result.
Yes, that's the definition of support. You have to consider a topogical space, a function defined in there with values in a vector space, and you define the support of the function as the complement of the nullity set of the function. And the nullity set is the biggest open set where the function is zero, that is, the interior of the set where the function is zero.
But anyhow, to talk about nullity and support, you have to fix a function, and this function has a domain. This domain has a topology. Depending on the topology you put on that domain you have different answers. But the correct definition is the one that first consider the nullity set and then defines the support as the complement of it.