Plot hole: In Episode 1-7: The Serene Squall, the Enterprise is at the edge of Federation space, and it is stated that it would take two days for a message to be received by Starfleet. Later in the episode, they have no problem establishing a real-time connection to Vulcan, which, according to canon, is only 16.5 light years away from Earth.
Other mistake: The episode begins with the Enterprise unable to contact in timely fashion the Federation, because there are 2 days of delay in communications, which puts Pike in a bind since apparently he needs authorization to get in non-Federation space (which kinda undermines the premise of the show). However, later in the episode, the Enterprise contacts the Vulcan facility on Omicron Lyrae without the slightest difficulty or delay.
Trivia: For anyone curious, the quote "I pretended to be somebody I wanted to be until finally I became that person. Or he became me" was said by Cary Grant - to Pelia, clearly.