The Extention of the Theory of Straight Lines and Second Degree Irregular Functions

The study recontextualizes previously examined Ireegular sequences and functions within the framework of linear theory, based on their observable derivative characteristics. The relocation is motivated by the consistent alignment between their derivatives, gradients and common differences-properties emblematic of linear functions, yet to be discovered in higher degree counterparts.

Subsequently, the investigation extends to second degree irregular functions, which exhibit cyclic traits analogous to those found in certain irregular and summation-defined functions. By analysing their derivatives, we gain deeper insight into their structural behaviour and potential classification.

The second degree counterparts display a hybrid mode, meaning they are formed by a combination of linear functions and curves. this hybrid natureis further asserted by the derivatives and are sketched. We also name them chain functions because of being made up ofmanystraight lines and curves or curves only. the first of these has been seen in the previous preprint article I rregular Functions: An Introduction (Mbele, 2025) that displayed a lot of joint straight lines as a single function.