This study considers the binary classification of functional data collected in the form of curves. In particular, we assume a situation in which the curves are highly mixed over the entire domain, so that the global discriminant analysis based on the entire domain is not effective. This study proposes an interval-based classification method for functional data: the informative intervals for classification are selected and used for separating the curves into two classes. The proposed method, called functional logistic regression with fused lasso penalty, combines the functional logistic regression as a classifier and the fused lasso for selecting discriminant segments. The proposed method automatically selects the most informative segments of functional data for classification by employing the fused lasso penalty and simultaneously classifies the data based on the selected segments using the functional logistic regression. The effectiveness of the proposed method is demonstrated with simulated and real data examples.