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?
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$\begingroup$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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