This is a very very good question because Jack Worthing's life, as he knows it, starts off with a tragedy: His loss at Victoria Station.
However, what Oscar Wilde does with that tragedy is to consider it a "triviality", and to make it part of the comedic storyline of the play. This is what adds a further touch of irony to an already ironic play, and what seals the Wildean style of making much out of nothing, and nothing out of much.
Jack Worthing's story basically goes like this: Miss Prism, his governess during his childhood, places her three-volume novel (in progress) in baby Jack's perambulator while she places the baby inside the handbag where her novel should have gone. That mistake alone raises eyebrows, considering that she left the baby and the handbag in a cloakroom at Victoria Station- "the Brighton line." Yet, happily for Jack, the man who finds him, Mr. Worthing, happens to be a rich man who raises Jack as his own child.
One can dilute the plot and find the tragedy of Jack's life to be more than Wilde admits. Lady Worthing, who is Jack's sister, admits at the end of the play that Jack's mother is an unfortunate woman who dies under unfortunate circumstances, always wondering where her baby is. If we imagine a mother trusting her baby to his sitter and then not seeing them again- it is indeed a very tragic situation.
Therefore, although the play is not aimed to be read as a tragedy, we can certainly see traces of it when Jack tells about his past. In fact, one almost feels sorry for Jack, for he has to find out about himself at an older age. How sad it may have been for him not to have a true identity. Yet,Wilde does not want us to read into the seriousness of the play. He wants us to look at what could be considered sad, and he wants us to laugh about it.