Why does Sir Sherlock Holmes get involved in the case in the first place?
The fact that Holmes takes up disappointments as challenges is evident from his dogged pursuit of the case even after the three lines of investigation did not yield any results. The three broken threads were the three lines of enquiry pursued by Sherlock Holmes in order to ascertain the identity of the man who was following Sir Henry Baskerville but they lead nowhere. The first one was to find out if Barrymore was the man stalking Sir Baskerville because the physical description of the man matched Barrymore's. However,this lead turned out to be cold because Barrymore had not ventured out of Baskerville which was proved by his personal receipt of a telegram sent to his residence. The second line of enquiry was to look for the cut-up Times paper from the waste baskets of 23 hotels near Charing Cross. This was to help Holmes find out the name of the hotel and subsequently, the name of the hotel guest who had sent the letter to Sir Baskerville. But that seemed to be a futile effort because it could not be found. Finally, after pursuing the cab driver and taking down the number of the cab, Holmes had sent for the cab driver to ascertain details of the stalker from the cab driver. This turned out to fall flat on him when the cab driver said that his passenger had confessed to being Sherlock Holmes, the detective. Thus, the man had succeeded in filching Holmes' identity and managed to escape without leaving any clue as to his identity or whereabouts. In spite of repeated failures, Holmes was obdurate in his quest to ascertain the identity of the stalker because it was a vital clue to the mystery that had been presented to him Dr Mortimer. This shows that failure was not a deterrent for him but an incentive to work harder.