Conserved mutual information for discrete and continuous variables in dilaton black hole

The information paradox of black holes has always been a hot topic at the forefront of theoretical physics. In this paper, we aim to understand the information paradox of the dilaton black hole from the perspective of the total correlation quantified by mutual information between the initial subsystems $A$ and $B$. We find that the dilaton gravity can redistribute initial mutual information for discrete and continuous variables between the subsystems $A$ and $B$, but cannot redistribute initial entanglement. Interestingly, the sum of physically accessible and inaccessible mutual information is equal to the initial mutual information. Therefore, physically accessible mutual information is transformed into physically inaccessible mutual information by the dilaton gravity of the black hole. From the perspective of mutual information, the information of black holes is conserved and not lost.