I would like to known if the following converse of the taylor's theorem is true:

Let $E$, $F$ Banach spaces, and $f:E\rightarrow F$ continuous. Suppose there are $k$ continuous functions $T_i: E \rightarrow L_i(E,F)$ such that for every $x \in E$ $$ f(x+h) = f(x) + T_1(x)h + T_2(x)h^2 + ... + T_k(x)h^k + r(x,h), $$ with $\lim_{h\to 0} \frac{r(x,h)}{\|h\|^k} = 0$. Then $f$ is $C^k$ and $f^{(k)}(x) = k! T_k(x)$

I found an analogue result with the same hypotesis on S. Dayal, "A converse of Taylor's Theorem on Banach Spaces", Proc. of Amercian Math. Soc., vol 68, but the conclusion is just that $f$ has Frechet derivative of order $k$.