What is the religiously prescribed time for Fajr prayer, and what is the ruling on not performing it within its time? Is it permissible to offer it as a make-up prayer due to negligence, old age, or not waking up early due to staying up late for work, or an illness that prevents waking up on time?

The time for Fajr prayer begins with the appearance of the true dawn (al-Fajr al-Sadiq) and ends with sunrise. The true dawn is the white light that spreads across the horizon. As for the false dawn (al-Fajr al-Kadhdhāb), no ruling is attached to it. The proof that the time ends with sunrise is the saying of the Prophet, peace and blessings be upon him: "Whoever catches one rak'ah of Fajr prayer before the sun rises has caught Fajr." It is not permissible to intentionally delay prayer beyond its prescribed time except for a legitimate Shar'i excuse. So, whoever sleeps through it or forgets it should pray it when he remembers it. Laziness and staying up late without necessity are not legitimate Shar'i excuses. A Muslim must be keen to perform prayers at their prescribed times.