Does the Euler-Lagrange functional actually have a functional derivative?

Question

It's well known that given a functional $$S(\boldsymbol q) = \int_a^b L(t,\boldsymbol q(t),\dot{\boldsymbol q}(t))\, \mathrm{d}t,$$ the directional (or otherwise known as the Gateaux) derivative $$\lim_{\epsilon \to 0}\frac{S(\boldsymbol q + \epsilon \boldsymbol \phi) - S(\boldsymbol q)}{\epsilon}$$ can be expressed as an integral $$\int_a^b \mathrm dt\,\phi(t)\cdot(L_x(t,q(t),\dot{q}(t))-\frac{\mathrm{d}}{\mathrm{d}t}L_v(t,q(t),\dot{q}(t)))$$ only when $\phi(a) = \phi(b)$, otherwise this integral expression doesn't work. But in general, if a functional $F$ has a functional derivative, then the Gateaux derivative $dF(q;\phi)$ of $F$ has to be expressible as an inner product of $\phi$ with the functional derivative of $F$ at $q$. In which case, does the functional derivative of $S$ actually exist? It seems not to, as it requires the condition that $\phi(a) = \phi(b)$.