I read that if you perform even one rakat of fajr before sunrise you have caught the prayer.
Is this the same if you are praying zuhr and asr starts during it?
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If you performed at least the first ruku' of a prayer, before the adhan of the next prayer or before the known ending time of the actual prayer, you've prayed on time.
This is based on the ahadith like:
Yahya related to me from Malik that he had heard that Abu Hurayra used to say, "Whoever catches the ruku has caught the sajda and whoever misses the recitation of the umm al-Qur'an has missed much good." (al-Muwatta')
This means performing ruku' is like completing the raka'a as the last thing in a rak'a (especially the first) is sujud.
Catch up here means before the end of the corresponding time for performing the prayer!
Yahya related to me from Malik from Zayd ibn Aslam from Ata ibn Yasar and from Busr ibn Said and from al-Araj-all of whom related it from Abu Hurayra - that the Messenger of Allah, may Allah bless him and grant him peace, said, "Whoever manages to do a raka of subh before the sun has risen has done subh in time, and whoever manages to do a raka of asr before the sun has set has done asr in time."
(al-Muawtta', Sahih al-Bukhari, Sahih Muslim and elsewehre)
The focus on subh (fajr) and 'asr here is because both prayers have an end time which is regulated by the sunrise or sunset and in case of fajr the adhan for the next prayer only starts hours later.
From the above ahadith it is clear that any prayer of which you've performed at least the first ruku' before the adhan or end of the time for the prayer is regarded as done on time. Whether it is valid or not or accepted or not is up to Allah the Almighty.