Studying the continuity of a real function in Tunisian secondary schools: from an intuitive approach to a formal definition

Abstract

Keywords:

Teaching, Continuity of real

functions, Formal definition,

praxeology.

Continuity, as a fundamental property of real functions, occupies an important

place in the teaching of Real Analysis. However, its rigorous definition remains

problematic.

This study focuses on the didactic approach to continuity in Tunisian curricula,

by analyzing the praxeological organizations proposed in the textbook for the 3rd

high school, Mathematics section class. The analysis is based on the theoretical

framework of the Anthropological Theory of Didactics (TAD). The results show

an evolution in the order of introduction of the notions of limit and continuity,

moving from limit-continuity to continuity-limit, with a formal definition of

continuity. This reform seems to aim for greater mathematical rigor, while

relying on graphical and kinematic intuitions. However, the study of

praxeological organizations reveals difficulties in the articulation between the

different registers (intuitive, graphical, and formal) around the definition of

continuity. Some proposed activities aim to progressively construct this formal

definition, but their implementation still seems problematic. These results raise

questions about the didactic conditions that promote a satisfactory understanding

of the continuity of real functions by students, particularly the balance between

intuitive, graphical and formal approaches. Areas for improvement are proposed

for a better articulation between the different aspects in teaching.