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?
Best Answer
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 $[0,\infty)$.