A　kernel　distribution　estimator　(KDE)　is　proposed　for　the　error　distribution　in　the　functional　data,　which　is　computed　from　the　residuals　of　the　B-spline　trajectories　over　all　the　measurements.　The　maximal　stochastic　process　between　the　KDE　and　the　error　distribution　is　shown　to　converge　to　a　Gaussian　process　with　mean　zero　and　specified　covariance　function　under　some　mild　conditions.　Thus,　a　simultaneous　confidence　band　(SCB)　is　constructed　for　the　error　distribution　based　on　the　KDE　in　the　dense　functional　data.　The　proposed　SCB　is　applicable　in　not　only　the　independent　functional　data　but　also　the　functional　time　series.　In　addition,　the　symmetric　test　is　proposed　for　the　error　distribution,　as　well　as　a　goodness-of-fit　test　for　mean　function　by　using　the　bootstrap　method.　Simulation　studies　examine　the　finite　sample　performance　of　the　SCB　and　show　the　bootstrap　method　performs　well　in　numerical　studies.　The　proposed　theory　is　illustrated　by　the　electroencephalogram　(EEG)　functional　data.　©　The　Author(s),　under　exclusive　licence　to　Springer　Science+Business　Media,　LLC,　part　of　Springer　Nature　2025.