Given that $T_j : \mathbb{R}^n \times \mathbb{R}^n\times \cdots \times \mathbb{R}^n\to \mathbb{R}^m$ $j$ times $\mathbb{R}^n$, such that $T_j$ is $j$-linear ,
and $f: \mathbb{R}^n \to \mathbb{R}^m$ which is $k-1$ differentiable.
given that for $a,h \in \mathbb{R}^n$ we have that $f(a+h)-\sum \limits_{j=0}^{k} T_j (h,\cdots ,h) = o(||h||^k)$
Does this imply $f$ is $k$ differentiable ?
My attempt : let $g(a) = D^{k-1} f(a)$ since $f$ is $k-1$ differentiable, i want to show that $g$ is differentiable and from this $f$ is $k$ differentiable.
$g(a+h) = D^{k-1} f(a+h) = D^{k-1} (f(a)+Df(a)h +\cdots +\alpha(h))$ such that $\alpha(h) = o(||h||^k)$
here were i get stuck
any help for proceed or another proof would really help me.