Let $L \in C^2$ and let $y \in C^1([a,b], \mathbb{R})$ be a stationary point of $J$. Then $\frac{\partial L}{\partial p}(x, y(x), y'(x))$ is continuously differentiable (i.e., $y$ automatically has the extra regularity needed to make $L_p$ $C^1$), and $y$ satisfies the **DuBois–Reymond equation** \begin{align*} \frac{\partial L}{\partial p}(x, y, y') &= \int_a^x \frac{\partial L}{\partial u}(t, y(t), y'(t))\, d\mathcal{L}^1(t) + c \end{align*} for some constant $c \in \mathbb{R}$, or equivalently, $\frac{d}{dx} L_p = L_u$ holds in the classical sense.