Let $a,b \in \mathbb{R}$ with $a < b$, let $L \in C^2([a,b]\times \mathbb{R}\times \mathbb{R};\mathbb{R})$, and write the variables of $L$ as $(x,u,v)$. Let $y \in C^2([a,b];\mathbb{R})$ satisfy the scalar Euler-Lagrange equation
Then for every $x_0 \in [a,b]$ there exists a relative neighbourhood $U$ of $(x_0,y(x_0),y'(x_0))$ in $[a,b]\times\mathbb{R}\times\mathbb{R}$ such that $\partial_{vv}L \neq 0$ on $U$. On this neighbourhood the function
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\begin{align*}
F: U \to \mathbb{R}, \quad F(x,u,v)=\frac{\partial_u L(x,u,v)-\partial_{xv}L(x,u,v)-v\,\partial_{uv}L(x,u,v)}{\partial_{vv}L(x,u,v)}
\end{align*}
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is continuous, and for every $x \in (a,b)$ such that $(x,y(x),y'(x)) \in U$ one has