In these notes on functionals, equation 4 transforms to equation 5 (on page 2). But I do not clearly understand where $\frac{1}{\epsilon}$ vanishes. The second line of page 3 tries to explain it, but I don't see how can $\frac{1}{\epsilon}$ be absorbed inconsequentially. http://julian.tau.ac.il/~bqs/functionals.pdf
The notes essentially want to do this, using the definition of integrals and I dont see where $\frac{1}{\epsilon}$ vanishes.
$$\lim_{\epsilon \rightarrow 0}\sum^{N}_{n=1}\epsilon(\frac{1}{\epsilon}\frac{\partial F}{\partial y_n})dy_n = \int_{a}^{b}\frac{\delta F}{\delta y(x)} \delta y(x) $$