Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, let $v \in \mathbb{R}^m$, and let $f: U \to \mathbb{R}^n$ be twice Fréchet differentiable at $a$. Write
for the [second derivative](/page/Second%20Derivative) of $f$ at $a$. Let $I \subset \mathbb{R}$ be an open interval with $0 \in I$ such that $a + tv \in U$ for every $t \in I$, and define the map
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\begin{align*}
g: I &\to \mathbb{R}^n, \qquad t \mapsto f(a + tv).
\end{align*}
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Then $g$ is twice differentiable at $0$ in the one-dimensional Fréchet sense, and