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Currently taking a higher level differential equations class and we're studying existence and uniqueness of solutions. Multiple times during proofs, my professor uses Lipschitz to say $$f(t,x)-f(t,y)\le L(x-y)$$ This concept makes sense to me as it only works if a function is Lipschitz. My question is, how can you tell if a function is Lipschitz, so that you can utilize this principle?

Different sites I have visited say that a function is Lipschitz if it satisfies the above inequality, which seems like circular logic to me.

One method I know of is if the function exists inside of a closed rectangle.

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There is no general criterion besides the definition, but there are some known results. The easiest one is to check whether $f$ is differentiable with bounded derivative, in which case it is Lipschitz with constant $||df||$, which is an immediate consequence of the mean value theorem (at least in convex regions). Often it is sufficient to know this locally.

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