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.