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Let $G$ be a finite, simple, and undirected graph of order $n$ and averagedegree $d$. Up to terms of smaller order, we characterize the minimal intervals$I$ containing $d$ that are guaranteed to contain some vertex degree. Inparticular, for $d_+\in \left(\sqrt{dn},n-1\right]$, we show the existence of avertex in $G$ of degree between$d_+-\left(\frac{(d_+-d)n}{n-d_++\sqrt{d_+^2-dn}}\right)$ and $d_+$.

Vertex degrees close to the average degree