I want to ask for a reference to study Variational Calculus from a formal point of view. What I mean is that many of the references that I've found are inside Physics books, and the authors do not care too much about mathematical rigour. For example, I think it's very important to make a distinction between a function $ f: \mathbb{R} \to \mathbb{R}$ and it's value on some point $y = f(x)$, however, those references usually speak loosely like "the function $y(x)$" and I really think that this isn't good.
Can someone point out then what's a good reference to learn variational calculus from a mathematically rigorous point of view?
Sorry if this question is silly, and thanks in advance!
I.M. Gelʹfand, S.V. Fomin "Calculus of variations".