Let \( n \) be a positive integer and \( f \) a real function defined on integers in the interval \( \left[ 1,n-1\right] \). Given a graph \( G \) with vertex set \( V \) and \( n \) non-isolated vertices, the degree-function index of \( G \) is defined as \( H_{f}\left( G\right) =\sum_{u\in V}f\left( d_{u}\right) \). It is our main objective in this paper to introduce the local value of a degree-function index \( H_{f} \) of a graph \( G \) at a vertex \( u \), which we denote by \( f_{G}\left( u\right) \). Intuitively, \( f_{G}\left( u\right) \) measures the contribution of vertex \( u \) in \( H_{f}\left( G\right) \). In this paper we initiate the study of its mathematical properties and address the problem of vertices with extremal local values in the zeroth-order general Randić index. In particular, for the first Zagreb index and the forgotten index, the problem of vertices with extremal local values is completely solved.