Time Warner Cable Inc. posted a 23 percent jump in its second-quarter earnings Thursday, boosted by strong results in its business services and higher revenue from residential users even as it continued to lose subscribers to its cable TV service. Like other cable companies, Time Warner has been working on growing its Web-based video service to better compete for customers who are increasingly turning to the Internet to watch movies and television shows. Cable TV providers have been losing subscribers to satellite TV companies and phone-company TV services for years. But in the second quarter of 2010 the overall pay-TV industry lost customers for the first time. The nation’s second-largest cable company behind Comcast Corp. said its net income rose to $420 million, or $1.24 per share, in the three months ended June 30, up from $342 million, or 95 cents per share, a year earlier. Earnings in the most recent quarter were boosted by a gain of 4 cents per share, but even without it they were better than what analysts had expected. Revenue rose 4 percent to $4.94 billion from $4.73 billion.