I am an undergraduate student in Mathematics and am interested in learning Calculus of Variations on my own, so I asked an older semester student if I could borrow his class notebook. Reviewing the first notes, I found a brief introduction to Functional Analysis, which includes the definition of functional and continuity of a functional. Later I saw the following example
Let $f\in C([a,b],\mathbb{R})$ and $J:C([a,b],\mathbb{R})\rightarrow\mathbb{R}$ the functional $J[x]=\int_a^b(f(t)+x^2(t))dt$.
In this example, he proved that this functional is continuous, using a trick to determine what is the proper delta for each given epsilon. I asked him what book his class was based on but he told me that his teacher never mentioned what book it was based on. I would like to review books in which this specific type of example comes, in which a functional is proposed that goes from a function space like $C([a,b],\mathbb{R})$ to real numbers and we are asked whether or not it is continuous, since that way I could become familiar with the reasoning behind the tricks to get the proper delta. Does anyone know of any books where this specific example comes? Or similar examples? Thanks in advance