Once I was trying to recall the definition what it means for $f$ to be a Lipshitz function on $E$:
$\exists c>0 \; \forall \delta>0 \; \forall x_1, x_2 \in E: \; (|x_1-x_2|<\delta \rightarrow |f(x_1)-f(x_2)| < c\delta)$.
And I erroneously formulated it as:
$\forall \delta>0 \; \exists c>0 \; \forall x_1, x_2 \in E: \; (|x_1-x_2|<\delta \rightarrow |f(x_1)-f(x_2)| < c\delta)$.
I quickly realized that it was not what I needed, but at the same time I wondered: what does mean the condition just formulated?
It's quite clear that we can just use $\varepsilon$ instead of $c\delta$, so it is simplified to $\forall \delta>0 \; \exists \varepsilon>0 \; \forall x_1, x_2 \in E: \; (|x_1-x_2|<\delta \rightarrow |f(x_1)-f(x_2)| < \varepsilon)$.
Then I started thinking how this condition can be formulated in "simple" words, like continuity, boundness, uniform continuity etc. I've spent more than hour and haven't come up with anything.
What I managed to infer by now:
It doesn't mean continuity even at single point, because for example the Dirichlet function satisfies it.
If $f$ is bounded, then it's true.
If $f$ is unbounded, it might be either true or false. For example $f(x)=x$ satisfies the condition with $\varepsilon=\delta$, and $f(x)=1/x$ with additional point $f(0)=0$ doesn't satisfy.
Can anyone say in commonly known terms, what does this condition mean? Does it have any natural meaning?
