Let (1) be a sequence of independent integer valued random variables. One says that for sequence (1) the local limit theorem is true in strong form if for each sequence which differs from (1) in only a finite number of terms relation (3) is fulfilled. We prove the following theorem: Condition (4) is necessary and sufficient that for the sequence (1) of random variables with distribution functions (2) the local limit theorem is true in strong form. If the distribution functions have the form (2a) a similar theorem is true.