Often, in introductory calculus, we study functions and their properties by drawing their graphs. In some sense, this mixes topology and analysis in mysterious (at least to me) ways. My question is about a finding a clear presentation of calculus that separates these two aspects.
For instance, the "intuitive" notion of a continuous function (analytic) is one where the graph of the function is connected (topological). Once we have formalised the analytic definition of continuity and defined the standard topology of the real plane, we can prove the formal statement:"The graph of a continuous function is connected but the converse is false".
Is it possible to characterise basic notions of calculus through statements of topology with some additional structure? In particular, I have the following concrete questions:
- Can we characterise the connected subsets of the coordinate plane that are graphs of continuous functions?
- What about graphs of differentiable functions? It appears I will need a formalism for differential topology that directly captures the notion of a "tangent line" without introducing the analytic notion of differentiation.
Note: I know the question is vague. I don't mind the use of modern mathematical formalisms like sheaves, diffeological spaces, nonstandard analysis and so on. I want clarity more than anything else.