The Lipschitz condition is a mathematical property that ensures a function does not oscillate too wildly, implying that it has a bounded rate of change. Specifically, a function is Lipschitz continuous if there exists a constant $L$ such that for all points $x$ and $y$ in its domain, the absolute difference between the function values is less than or equal to $L$ times the distance between $x$ and $y$, or mathematically, $$|f(x) - f(y)| \leq L |x - y|$$. This concept is crucial in numerical methods as it guarantees the uniqueness and stability of solutions for differential equations, connecting strongly to methods like Euler's Method and Euler-Maruyama Method.
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