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?

  • $\begingroup$ This has nothing to deal with coninuity. Functions satisfying this condition are simply locally bounded functions. $\endgroup$ – Crostul Dec 5 '20 at 22:15
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    $\begingroup$ 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 $f$. $\endgroup$ – Sangchul Lee Dec 5 '20 at 22:16

Hölder continuity is defined as :

$$\forall x,y \in \mathbb R^d , |f(x)-f(y)| \leq C|x-y|^\alpha$$

Maybe if you set $\alpha=0$ you get what you are working with?


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