Morse Theory
Lipschitz continuity is a strong form of uniform continuity that ensures a function does not change too rapidly; specifically, if a function $f: X \to Y$ is Lipschitz continuous, there exists a constant $K \geq 0$ such that for all points $x_1, x_2 \in X$, the inequality $|f(x_1) - f(x_2)| \leq K |x_1 - x_2|$ holds. This property is significant as it guarantees the boundedness of the function's derivative and plays a crucial role in understanding the behavior of smooth functions and their stability under perturbations.
congrats on reading the definition of Lipschitz Continuity. now let's actually learn it.