The article defines stability for centrality measures in weighted graphs. Degree, closeness, and eigenvector centrality are stable, but betweenness centrality is not. An alternative definition for stable betweenness centrality is introduced. Continuity is a less strict version of stability. Betweenness centrality is not only unstable but also discontinuous. In practice, stable centrality measures are more robust against noisy data compared to unstable ones. Stable betweenness centrality shows resilience against noise while maintaining its centrality concept.