A function $f:\mathbb{R} \to\mathbb{R}$ is continuous at a point $c$ if, for every $\varepsilon >0$ there exists a $\delta > 0$ such that $|x-c| < \delta$ implies $|f(x)-f(c)|<\varepsilon$. For a given continuous function $f$, consider the function $g:\mathbb{R}\times\mathbb{R}^+ \to\mathbb{R}^+\cup\{\infty\}$ given by $$g(x, \varepsilon) = \sup\{\delta : |x-y| <\delta \Rightarrow |f(x)-f(y)|<\varepsilon\}.$$
Does this function have a name? Are there restrictions we can put on $f$ to make it well-behaved? I don't think I have the real analysis chops to investigate this in much depth on my own, but if someone has already studied it I would like to read about it.