# Is there a name for the “opposite” of continuity?

Recently I was working on a proof where I eventually wanted to show that for a continuous function $$f$$ we have the following property at $$x$$:

For all $$\delta > 0$$ there exists some $$\epsilon > 0$$ such that for every $$x' \in N_\delta(x)$$:

$$|f(x') - f(x)| \leq \epsilon$$

This statement looks superficially like a statement about the continuity of $$f$$, but it seems to reduce to a claim about $$f$$ being bounded on a neighborhood at $$x$$:

$$\sup_{x' \in N_\delta(x)} f(x') < \infty$$

Is there more to this "opposite" definition of continuity? Does it have another name? Is it really just that $$f$$ is bounded on a neighborhood?

• This has nothing to deal with coninuity. Functions satisfying this condition are simply locally bounded functions. – Crostul Dec 5 '20 at 22:15
• The proposed property need not hold for any continuous function. It is in general much stronger than local boundedness. For example, $f:(0,\infty)\to\mathbb{R}$ with $f(x)=1/x$ is continuous (and hence locally bounded), but it does not satisfy OP's property. Another line of example comes when the domain is a bounded metric space, in which OP's property is the same as boundedness of $f$. – Sangchul Lee Dec 5 '20 at 22:16

$$\forall x,y \in \mathbb R^d , |f(x)-f(y)| \leq C|x-y|^\alpha$$
Maybe if you set $$\alpha=0$$ you get what you are working with?