The paper presents a stability result for persistence diagrams, which are used to encode topological information of a function on a topological space. The main result shows that small changes in the function lead to small changes in the persistence diagram. This stability is proven for both the Hausdorff and bottleneck distances. The result is applied to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes. The stability of persistence diagrams is crucial for analyzing topological features of data, as it ensures that small perturbations in the data do not significantly affect the topological structure. The paper also discusses the implications of this result for various applications, including the estimation of homology groups from point samples and the comparison of geometric shapes based on their persistence diagrams. The results are supported by theoretical proofs and examples, demonstrating the robustness and practical relevance of persistence diagrams in topological data analysis.The paper presents a stability result for persistence diagrams, which are used to encode topological information of a function on a topological space. The main result shows that small changes in the function lead to small changes in the persistence diagram. This stability is proven for both the Hausdorff and bottleneck distances. The result is applied to estimating the homology of sets in a metric space and to comparing and classifying geometric shapes. The stability of persistence diagrams is crucial for analyzing topological features of data, as it ensures that small perturbations in the data do not significantly affect the topological structure. The paper also discusses the implications of this result for various applications, including the estimation of homology groups from point samples and the comparison of geometric shapes based on their persistence diagrams. The results are supported by theoretical proofs and examples, demonstrating the robustness and practical relevance of persistence diagrams in topological data analysis.