Undergraduate students' understanding of function continuity has not been explored broadly in previous research. The relevant findings in the literature are predominantly concerned with calculus students' understanding and misconceptions of continuity. Many of these misunderstandings are tied to the relationships which continuity has with limits and differentiability. This multiple-case study explores how, if at all, an introductory real analysis course impacts undergraduate students' understanding of function continuity and its connections to the notions of limits and differentiability. We embed our findings within the theoretical framework of Tall's three worlds of mathematics, namely, the embodied, symbolic, and formal worlds. The cases in this study are seven undergraduate students who enrolled in an introductory real analysis course during the Fall 2022 semester. Each participant took a pre-test to assess their comprehension of function continuity and its relations with limits and differentiability before the topic was covered in the course, as well as a post-test that measured their understanding of the subject after learning about it in real analysis. Finally, six of the seven cases engaged in an individual, audio-recorded oral interview with questions designed to triangulate, enrich, and clarify the data collected from the written instruments. The pre- and post-tests were scored numerically, with each subject receiving pre- and post-embodied, symbolic, and formal scores, while the qualitative data collected from the interviews were coded with respect to the three worlds of mathematics and analyzed via constant comparison. In all, we found that the students' perspectives on function continuity generally shifted from an embodied viewpoint to a more symbolic and formal outlook, though several of the misconceptions held by calculus students, per the existing literature, were possessed by the real analysis students as well, even after the material was covered in class. In particular, several cases conflated continuity with the concepts of connectedness and domain. Moreover, the ability of a student to state the correct relationship between function continuity and limits did not guarantee that the student could apply this knowledge; the same was true about the connections between function continuity and differentiability. Of high interest was the strength of the cases' concept images regarding continuity and the ways in which inadequate concept images permeated their understanding of the topic throughout the study. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]