The first equation indicates that for any two points in R^N, the maximum norm of the difference in gradient is less than a constant times the distance between the points.
The keyword to google for is just "Lipschitz".
So, f(x) = x^2 is not Lipschitz-continuous (because the slope gets arbitrarily large), but something like f(x) = sin(x) is Lipschitz-continuous because the slope never exceeds some upper bound.
Funny how trying to write down the question gives the brain the kick it needs sometimes :)
In some situations, it's enough to prove an equation is Lipschitz-continuous on a range. Example, y=x^2 is lipschitz continuous on x=[0,1].