Continuity and differenciability

In Tall’s formulations, the graphical representation of the differential function, when enlarged to a determined portion, looks like, locally, a segment of a straight line. Afterwards, the researcher formulated the notion of cognitive roots “local straightness”, which is based on the perception that tiny part of certain graph under high magnification eventually looks virtually straight (TALL, 1989). This notion would be appropriate to the development of the concept of derivative because “it allows the gradient function to be seen as the changing gradient of the graph itself” (TALL, 1993, p. 2). By the notion of local straightness, it would be possible to stimulate the student’s imagination to conceive how a graphic representation of a continuous and non-differentiable function at the points of the domain would be. A characteristic of this representation would be the following: it should keep the “beak”, not mattering how much this function is enlarged. The blancmange function would be an example of this fact, because of the way it is defined.