Non-differentiability implies non-Lipschitz continuity

In a simple one-dimensional framework, it is known that the differentiability of a function (with bounded derivative) on an interval implies its Lipschitz continuity on that interval. However, non-differentiability does not implies non-Lipschitz continuity as shown by the function $f:x\to |x|$. Still, there are functions that are not differentiable at a point and this argument is used to say that this same function is not Lipschitz-continuous at that point as, for example, $f:x\to \sqrt{x}$ at $x=0$ (we say that the slope of the function at that point is "vertical"). So my question is: is there a theorem telling us that for some functions (to be characterized), their non-differentiability implies their non-Lipschitz continuity? Or is this result obvious from the definition of Lipschitz continuity?