<p>In this paper, we introduce a new class of contractive definitions known as convex Meir-Keeler-Ćirić-Matkowski contractive mappings. We establish several fixed point theorems under this new condition, allowing for both continuity and discontinuity at the fixed points. Our results not only encompass all previously known findings in this domain but also offer new insights into the continuity of contractive mappings at their fixed points. As an application of our theorem, we demonstrate the existence and uniqueness of solutions to a functional equation in the Lipschitz space. The functional equation we consider broadly encompasses various functional equations, including those recently studied for analyzing the two-choice behavior of the paradise fish and for solving models involving two prey species and one predator.</p>