Follow up to this post. Is my attempt at a proof of the theorem below correct?
Definition: let $C^1[a,b]$ be the collection of differentiable functions $[a,b]\to\mathbb{R}$ with continuous derivative. The space forms a vector space with the natural operations inherited from $\mathbb{R}$. Furthermore, the map $||\cdot||_1:C^1[a,b]\to\mathbb{R}$ defined by $$y\mapsto ||y||_1 := \sup_{a\le x\le b} |y(x)| + \sup_{a\le x\le b}|y'(x)|$$ is a norm on $C^1[a,b]$, providing it with the structure of a normed vector space, and thus of a topological space.
Definition: a functional $J:C^1[a,b]\to\mathbb{R}$ is continuous if it is continuous with respect to the topology provided on $C^1[a,b]$ by its norm, and the standard topology of $\mathbb{R}$. We say $J$ is differentiable at $y\in C^1[a,b]$ if its Fréchet derivative exists at that point, in which case we denote it by $\delta_y J$.
Theorem: for a fixed $C^2$ function $F:\mathbb{R}^3\to\mathbb{R}$ we define the functional $$J:C^1[a,b]\to\mathbb{R}:y\mapsto\int_b^aF(x,y,y')\ dx.$$ Then $J$ is differentiable at any $y\in C^1[a,b]$ with $$\delta_yJ:h\mapsto\int_b^a\left[F_y(x,y,y')h+F_{y'}(x,y,y')h'\right]\ dx.$$
Proof (my attempt): by Taylor's Theorem we may write $F(t,y+h,y'+h')$ as \begin{split} & \ F(t,y,y')+\sum_{|\alpha|=1}\partial_\alpha F(t,y,y')\mathbf{h}^\alpha + R_{\mathbf{a},k}(\mathbf{h})\\ = & \ F(t,y,y')+\partial_2 F(t,y,y')h + \partial_3 F(t,y,y')h' + R_{\mathbf{a},k}(\mathbf{h}) \end{split} for $\mathbf{a}=(t,y,y')$ and $\mathbf{h}=(0,h,h')$. We thus have \begin{split} J(y+h)-J(y) & = \int_a^b\Bigg[F(t,y+h,y'+h')-F(t,y,y')\Bigg]\ dt\\ & = \int_a^b\Bigg[\partial_2 F(t,y,y')h + \partial_3 F(t,y,y')h'\Bigg]\ dt + \int_a^bR_{\mathbf{a},k}(\mathbf{h})\ dt \end{split} and it remains to be shown that $$\lim_{\|h\|\to 0}\frac{\int_a^bR_{\mathbf{a},k}(\mathbf{h})\ dt}{\|h\|}\to 0$$ where the remainder $R_{\mathbf{a},k}(\mathbf{h})$ takes the form $$\sum_{|\alpha|=2}\frac{\partial_\alpha F(t,y+c_{\alpha,t}h,y'+c_{\alpha,t}h')}{\alpha!}\mathbf{h}^\alpha.$$ Letting $$r=\sup\bigg\{\|\left(t,y+c_{\alpha,t}h,y'+c_{\alpha,t}h'\right)\| : t\in[a,b]\bigg\}$$ then, for any $\alpha$ with $|\alpha|=2$, the function $\partial_\alpha F$ is continuous on the closed ball $B_r(0)$ (of radius $r$, centered at $0$) and thus attains its maximum $M_\alpha$ somewhere on such set. If we let $M=\sup_{\alpha}M_\alpha$, then \begin{split} \frac{1}{\|h\|}\int_a^bR_{\mathbf{a},k}(\mathbf{h})\ dt & \le \frac{1}{\|h\|}\int_a^b\sum_{|\alpha|=2}M\mathbf{h}^\alpha\ dt\\ & = \frac{M}{\|h\|}\int_a^b\bigg[h^2+2hh'+(h')^2\bigg]\ dt\\ & \le \frac{M}{\|h\|}\int_a^b\|h\|^2\ dt\\ & = M(b-a)\|h\| \end{split} which tends to $0$ as $\|h\|\to 0$.