I am an outsider for this field of mathematical analysis. But to analyse a problem of Control Systems, which is my area of interest, I need to know this.

I learned that absolutely continuous functions are also differentiable almost everywhere. On the other hand, Lipschitz continuity ensures bounded derivative of the function a.e. I am wondering whether there is any link between the derivative being bounded and the function being absolutely continuous. Or, is there any sufficient condition to be imposed over the absolute continuity to ascertain that the derivative of the function will be bounded?


If a continuously differentiable function has a bounded derivative, then it is absolutely continuous. The inverse is not true, as shown by the function $f(x)=\sqrt{x}$ on $[0,\infty)$.

  • $\begingroup$ thanks for your answer. Hence, can it be concluded that Lipschitz continuity is the weakest condition to be imposed over absolute continuity to ensure bounded derivative of the function? Is my understanding right? $\endgroup$ – neelarnab Feb 23 '15 at 12:45
  • $\begingroup$ All differentiable functions with a bounded derivative are lipschitz. All lipschitz functions are absolutely continuous. All absolutely continuous functions are continuous. None of these inclusions can be inverted. So I would say that Lipschitz continuity is a stronger condition, because it works on more functions than bounded derivatives. It is easier to chech bounded derivatives. $\endgroup$ – 5xum Feb 23 '15 at 12:48

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