If the derivative of a function is lipschitz,,,does it mean that the function itself is also lipschitz? Any proof for that?



Lipschitz condition on the first derivative of a function?
No. $f(x)=x^2$ on the whole real line is not Lipschitz, but the derivative $f'(x)=2x$ is.
However, if the function is defined on a bounded interval, then the statement is true.
Indeed, if $f'$ is Lipschitz, then it is continuous and thus bounded, and a function
with bounded derivative is Lipschitz.


