  Given a filtration function on a finite simplicial complex, stability theorem
of persistent homology states that the corresponding barcode is continuous with
respect to changes in the filtration function. However, due to the discrete
setting of simplicial complexes, the critical simplices terminating matched
bars cannot change continuously for arbitrary perturbations of filtration
functions. In this paper we provide a sufficient condition for rigidity of a
terminal simplex, i.e., a condition on $\epsilon > 0$ implying that the
terminal simplex of a homology class or a bar in persistent homology remains
constant through $\epsilon$-perturbations of filtration function. The condition
for a homology class or a bar in dimension n depends on the barcodes in
dimensions n and n+1.
