Consider the family of functions on $[a,b]$ given by $y(x; \epsilon) : = u(x) + \epsilon \eta (x)$, where the functions $\eta = \eta (x)$ are twice continuously differentiable and satisfy $\eta(a) = \eta(b) = 0$. Here $\epsilon$ is a small real parameter,..
What does this mean? What is the meaning and use of the "small real parameter". I'm currently doing a bit of self-studying, nothing too serious but I am interested in this stuff, but of course it is tough without guidance, and the biggest stumbling block for me is the notation and the jargon.
I've never really encountered the notation $y(x;\epsilon)$, what is the purpose of the semi-colon?
Source: http://www.macs.hw.ac.uk/~simonm/mechanics.pdf , top of the fourth page. Beginning of a proof of the Euler-Lagrange equation.