Lipschitz continuity is a property of a function that guarantees a bound on how much the function can change in relation to changes in its input. Specifically, a function $f(x)$ is Lipschitz continuous on an interval if there exists a constant $L$ such that for any two points $x_1$ and $x_2$ in that interval, the absolute difference in function values is at most $L$ times the absolute difference in inputs: $$|f(x_1) - f(x_2)| \leq L |x_1 - x_2|$$. This property is crucial in root-finding methods as it implies the function does not oscillate wildly, ensuring stability and convergence of these numerical techniques.
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