In calculus of variation, they would often claim that any well-behaved function $\tilde{y}(x)$ can be expressed as $y(x)+\epsilon g(x)$, where $y(x)$ is our stationary function. I cannot find the proof of this.
After a bit of research, I feel like this is related to theorem 9.5 of the book "a second course in real functions". This theorem states that every real valued function is the difference of two Darboux continuous functions. But I don't know to show that we can fix one of the functions in the sum.