This paper is a continuation of an earlier discussion in this journal about adhering to principles of mathematics while presenting function graphs in physics. As in the previous paper, the importance of the vertical line test was examined, this paper delves more in-depth, and it pinpoints a need for presenting graphs with a continuous rate of change in cases when phenomenon requires such condition to be met. Graph analysis is frequently researched in science and mathematics education. Students' misconceptions are highlighted, and methods of improvement suggested. However, graphs might inadvertently produce unrealistic interpretations if they do not adhere to principles of continuous and differentiable algebraic functions. A survey of several physics resources has revealed that algebraic conditions of continuity and differentiability are often neglected, which might ultimately question the graphs' truthfulness and confuse the learners. Presenting such graphs to students can also undermine the merit of algebraic methods used in learning physics and teaching these methods in advanced mathematics courses.