Taking breaths during Shofar blowing:
A. Within a single sound:
[The minimum required measurement of the Tekiah blow must be done in one breath.] The [minimum required] amount of the Yevavos [i.e. Terumatin] of the Teruos in the sound of Terua, as well as the three Shevarim in the sound of Shevarim, must be done in one breath. If one did not blow them in one breath, he does not fulfill his obligation. [Thus if he took a breath in the middle of the three/nine Terumatin of the Teruah or in middle of the three Shevarim he must restart the blowing of that Teruah or Shevarim. The above however is only if one did not blow a Kosher Tekiah or Kosher Teruah or Kosher Shevarim in either the sound before the breath or the sound after the breath, in which case they do not accumulate to make up minimum required measurement. If, however, one of the sounds were valid, either the one before the breath or the one after the breath, then this is considered that one blew an extra sound in middle of the set of which at times the ruling is that one may continue with the blows, and at times he must repeat from the first Tekiah of that set. See Halacha 12 regarding one who blew an extra sound for the full details of this subject [see 12D in particular].
If one did not take a breath in middle of the sound but made a silent break in middle and then continued what is the law?
Some Poskim rule that if the silent break in a sound is equivalent to the amount of time it takes for one to take a breath and begin to blow then it has the same law as one who took a breath in between. It requires further analysis as to the opinion of Admur in this matter.
The Tekiah: Seemingly by the Tekiah even the slightest break of silence in the sound invalidates the sound even if one did not take a breath, unless one of the parts of the sound, before or after the break of silence, contained the minimum measurement of length of a Tekiah.
 See Admur 590:8 and 15-17
 Pashut, as otherwise it is not considered a Tekia, and seemingly this is the reason that it was not explicitly mentioned in Poskim here
 Admur 590:8; Michaber ibid; Poskim ibid
The reason: As the three Shevarim is considered the Teruah of that set. [Admur ibid; Tur ibid; Kaf Hachaim 590:27]
 Admur 590:8; Michaber 590:4; Tur 590; Ramban Rosh Hashanah 34a; Rosh R”H 4:11; Rabbeinu Tam
 Admur ibid; Kneses Hagedola 590:7; Elya Zuta 590:7; Implication of Beis Yosef 590; Abudarham; Rabbeinu Yerucham Nesiv 6:2; Orchos Chaim; Mamar MordechaI 590:8; Nivei Shalom 590:5; Biur Hagr”a 590; Chayeh Adam 142:8; M”B 590:16; Kaf Hachaim 590:28
The reason: As the Torah stated that one must blow a single Teruah and not two Teruos. [Admur ibid; Levush 590:4; Ramban ibid; Kaf Hachaim 590:28]
Other opinions: Some Poskim rule that Bedieved if one took a breath within a single sound, he still fulfills his obligation. [M”A 590:7; Taz 590:5; Derech HaChaim 7; Mateh Efraim 590:12; See Kaf Hachaim ibid]
 Mateh Efraim 590:25; Piskeiy Teshuvos 590:5 and 9 footnote 44
 Admur never mentions that a silent break has the same status as a breath. On the contrary from his wording in different Halachas it is implied that it is all one sound. [See 590:4; 590:16 and 590:17] So is also implied from M”B 590:34; Vetzaruch Iyun.
 See Piskeiy Teshuvos 590 footnote 21 and so seems logical as if it is valid to make a break of silence within a Tekiah then what would differentiate a Tekiah from a Shevarim! Hence, one must say that the minimum length of the Tekiah, which is Lechatchila 9 Terumatin long in Tashat and Tarat and 18 in Tashrat, must be done without any break in between.