There is a particular detail about Lipschitz continuity about which I am unsure. I would define Lipschitz continuity as follows. For simplcity I assume a real-valued function on the real line. Is the following correct?
There exists positive constants a and b such both $\vert x-x_0\vert < a$ AND $ \vert f(x) - f(x_0)\vert < b \vert x-x_0 \vert$ are true. Note that I do NOT require: Given $b$ there exists $a$ etc etc, or vice versa. I merely require that the constants $a$ and $b$ exist.
This does not exclude the if $b$ then $a$ .... relationship, it merely is not required. The inverse - if $a$ then $b$ - is also allowed, which gives a lot of flexibility.
Am I missing something?