Tracing students understanding and misconceptions of continuity and differentiability through multi representations of mathematical problem

Purpose: This study investigates undergraduate students’ understanding and misconceptions concerning continuity and differentiability across multiple mathematical representations. It specifically examines whether misconceptions remain stable or shift when concepts are presented in symbolic, graphical, and contextual forms. Method: A qualitative exploratory case study was conducted involving 83 undergraduate mathematics education students at a leading public university in Indonesia. Data were collected using 11 validated and reliable multiple-choice items designed to elicit reasoning across different representations. Students’ written responses were analyzed to identify patterns of understanding and error. In-depth interviews with selected participants were conducted to clarify underlying reasoning processes. A collective case study framework was employed, treating each representation as a micro case to enable systematic cross-representation comparison. Findings: The results indicate that misconceptions were widespread and often consistent across representations. Common errors included interpreting graphical smoothness as proof of differentiability, assuming differentiability without verifying continuity, and conflating the existence of a limit with the function’s value. Many students relied on procedural manipulation, visual intuition, or contextual familiarity rather than formal limit-based reasoning, revealing fragmented conceptual understanding. Significance: The findings highlight the need for instructional approaches that explicitly integrate multiple representations and strengthen connections between intuitive reasoning and formal definitions. By addressing persistent misconceptions across representations, calculus instruction can better support the development of coherent and conceptually grounded mathematical understanding.