Answer to Question #137631 in Differential Equations for Gani

Question

Every continuous function ... Satisfy a Lipschitz consution

Answer 1

$f(x)=x^2.$ Its a continuous function. Now if it satisfies Lipschitz condition, then $|f(x)-f(y)|\leq k|x-y|$ for some $k>0.$ We take $x=2k, y=0$ . Then $|f(x)-f(y)|=|x^2-y^2|=|x+y||x-y|=2k|x-y|>k|x-y|$ . Hence doesn't stisfy Lipschitz condition.