We　investigate　the　existence　of　a　global　classical　solution　to　the　Goursat　problem　for　linearly　degenerate　quasilinear　hyperbolic　systems　of　diagonal　form.　As　the　result　in　[A.　Bressan,　Contractive　metrics　for　nonlinear　hyperbolic　systems,　Indiana　Univ.　Math.　J.　37　(1988)　409-421]　suggests　that　one　may　achieve　global　smoothness　even　if　the　C1　norm　of　the　initial　data　is　large,　we　prove　that,　if　the　C1　norm　and　the　BV　norm　of　the　boundary　data　are　bounded　but　possibly　large,　then　the　solution　remains　C1　globally　in　time.　Applications　include　the　equation　of　time-like　extremal　surfaces　in　Minkowski　space　R1+(1+n)　and　the　one-dimensional　Chaplygin　gas　equations.