The discussion revolves around the definitions of smooth curves and piecewise smooth curves in the context of continuous functions mapping an interval [a,b] into real numbers.
The discussion is active, with participants providing definitions and raising questions about the nuances of these concepts. There is no explicit consensus, but various interpretations and clarifications are being explored.
Participants are examining the implications of smoothness and continuity, particularly in relation to the existence of derivatives and the behavior of curves over specified intervals.
Klockan3 said:A smooth curve is a curve for which all of its derivatives exists, which gives that all derivatives must be continuous and smooth. A piecewise smooth curve is a curve for which there exists a set of intervals where the curve is smooth on each of those intervals. As an example, a curve might not be smooth on [a,b] but it is still piecewise smooth if it is smooth on both [a,c] and (c,b].