Let $E_1,...,E_n$ and $F$ be normed vector spaces and let $\Omega \subset \prod E_k$ be open.
Let $f: \Omega \to F$ and let $a \in \Omega$
Are the following conditions sufficient for $f$ to be differentiable at $a$ (Fréchet-wise)?
The partial derivative maps of $f$ exist on $\Omega$.
The partial derivative maps of $f$ are continuous at $a$.
Why am I asking the question
I read somewhere a theorem which stated that these two conditions plus the continuity of $f$ on $\Omega$ are sufficient for $f$ to be differentiable at $a$. However, it appears to me that the continuity of $f$ is used nowhere in the proof.
I apologize for not presenting the proof: the source is not available online, and the proof is too long to write it down.
Thus, I'm just expecting an experienced person to confirm to me whether or not the continuity is needed; I am not looking for a proof, but just a hint for why the continuity of $f$ is needed (in case it is).
Say $a=0$ and $f(a)=0$, for simplicity. If one of the derivatives, say $\partial_{x_1}f$, is continuous in a neighborhood of $0$ then $f$ is continuous also, because $$ f(x_1\ldots x_n)=\int_0^{x_1}\frac{\partial f}{\partial x_1}(y_1, x_2\ldots x_n)\, dy_1.$$