The huge amount of available data nowadays is a challenge for kernel-based machine learning algorithms like SVMs with respect to runtime and storage capacities. Local approaches might help to relieve these issues and to improve statistical accuracy. It has already been shown that these local approaches are consistent and robust in a basic sense. This article refines the analysis of robustness properties towards the so-called influence function which expresses the differentiability of the learning method: We show that there is a differentiable dependency of our locally learned predictor on the underlying distribution. The assumptions of the proven theorems can be verified without knowing anything about this distribution. This makes the results interesting also from an applied point of view.