Variation and functional derivative

Question

Let $F$ be a functional on a certain space of functions (actually I don't know what's necessary to define functional derivative). Is the following equality true for any small variation $\delta f(x)$ ? \begin{align*} \delta F[f(x)]:&=F[f(x)+\delta f(x)]-F[f(x)]\\ &=\int dx\,\delta f(x) \frac{\delta F}{\delta f}(x)+\mathcal{O}(\|\delta f\|^2) \end{align*} If the variation is denoted as $\epsilon \phi(x)$ with $\epsilon\to 0$, it's easy to see. However, I don't know how to deal with an arbitrary variation.