Vertex degrees close to the average degree

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