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Define the oscillation of a function at a point $x$ to be (for an open interval $I$):

$$\omega_f(x)=\inf_{x\in I}\sup_{s,t\in I}|f(t)-f(s)|$$

I am a bit confused about the definition above. How am I supposed to interpret the infimum of the supremum of something? The smallest maximum distance between two function values? If I have the maximum distance between two function values, what is there to choose an infimum for? Can someone please clarify?

Thanks.

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1 Answer

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Start with a point $x$.

Let $I$ be an open interval containing $x$. The oscillation of $f$ on $I$ is the quantity $\displaystyle \sup_{s,t \in I} |f(t) - f(s)|$.

For all such $I$ containing $x$ you get a value for the oscillation of $f$ on $I$. The oscillation of $f$ at the point $x$ is the infimum of all such values.

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