At the moment I am studying a chapter about Sobolev spaces in a book on partial differential equations authored by Evans. I have a question about continuity of a function in a broad sense. More specifically how to show that a given function is continuous. A function could be continuous in different ways. Such as Holder, Lipschitz or uniform continuous. We could also define a function as being continuous in a weak or strong sense.
I am aware of the basic definition of continuity. A small change in the input should lead to a small change in the output. No jumps should occur in a graph of a function. I know a little why it is important. For instance: Picard-Lindelof theorem.
I asked my self how I could know that a function is continuous or not was by just plotting it in a plotting tool and evaluate the graph. No jumps in the graph means that it is continuous. However, if I would do this this would lead to a few confusions. Such as how can I see that it is Holder or Lipschitz continous? What is the Lipschitz constant? If this would work so well why do we need use a weak formulation any way? Why do we just not put the non-weak formulated function in a program, plot it and make our conclusions?
About what I said about weak formulation. I do know why we use a weak formulation (according to the explanation of Wikipedia and the book that I described earlier). However, I am unable to connect all the dots.
EDIT: I am also aware of the epsilon-delta definition.
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