One of the sounds blown with the Shofar on Rosh Hashanah is the Shebarim sound, which consists of three medium-length blasts. The majority view among the Rishonim (Medieval Halachic scholars) is that the person blowing the Shofar must blow the Shebarim in a single breath; he must not take a breath in between any of the three blasts. If he does take a breath in the middle of the Shebarim, then he and the congregation have not fulfilled the obligation of Shofar blowing on Rosh Hashanah.
Another sound that is blown with the Shofar is Shebarim-Teru’a, meaning, a Shebarim which is immediately followed by a Teru’a. The Shulhan Aruch (Orah Haim 590:4) records a debate as to whether or not one should take a breath in between the Shebarim and Teru’a. One opinion requires blowing the Shebarim and Teru’a in a single breath, while others maintain that one should specifically make a point to take a breath in between the two sounds. The Shulhan Aruch concludes that a Yereh Shamayim (God-fearing individual) should seek to satisfy both opinions. This is done by sounding the Shebarim-Teru’a in a single breath during the first set of Shofar sounds, which are blown before Musaf (known as the "Teki’ot De’meyushab"), and then sounding the Shebarim-Teru’a with a breath in between when blowing the Shofar during Musaf. This is, indeed, the prevalent custom. In fact, some Mahzorim write "Shebarim-Teru’a" in the first set of Shofar blasts with a hyphen in between the two words, indicating that they should be sounded in a single breath.
If the one blowing the Shofar is unable to sound the Shebarim-Teru’a in a single breath, and he takes a breath in between the Shebarim and Teru’a, the congregation nvertheless fulfills their obligation. This is the ruling of Hacham Ovadia Yosef. Preferably, however, the congregation should ensure to appoint somebody who is capable of sounding the Shebarim-Teur’a in a single breath during the first set of Shofar blasts.
Summary: The Shebarim sound must be blown in a single breath. During the first set of Shofar blasts blown before Musaf, one should blow the Shebarim-Teru’a sound in a single breath. During the Shofar blasts sounded as part of the Musaf service, one should pause to take a breath in between the Shebarim and Teru’a. If one pauses to take a breath in between the Shebarim and Teru’a during the first set of Shofar blasts, he and the congregation nevertheless fulfill their obligation.