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Classical authorities pick up the discussions very well and also give variant meanings to the root word D-L-K one of which is a reference to sunset.

I am not sure whether I would accept the proposition that one prays (salaat) continuously from the Zenith, all throughout the day and till the darkness of the night.

The root word D-L-K is also reference to sun rise
( http://www.studyquran.org/LaneLexicon/Volume3/00000072.pdf ):

It is  said in [one of the works entitled] the "Nawadir el-Aarab," that dalakatialshshamsu signifies The sun became high

Using that meaning and regard 17.78, people not sleep in mid-day 'aqimi alssalata' from sun rise (wake in the morning) till the darkness of the night (sleep in the night).

People sleep in mid-day 'aqimi alssalata' in the first end of the day, sleep in mid-day, then 'aqimi alssalata' in the second end of the day and zulafan mina allayli (till sleep in the night).

Hud (The Prophet Hud)

Waaqimi alssalata tarafayi alnnahari wazulafan mina allayli inna alhasanati yuthhibna alssayyiati thalika thikra lilththakireena

11:114 (Y. Ali) And establish regular prayers at the two ends of the day and at the approaches of the night: For those things, that are good remove those that are evil: Be that the word of remembrance to those who remember (their Lord):

'Aqimi alssalata' is a long life, and live not include sleep,

Maryam (Mary)

WajaAAalanee mubarakan ayna ma kuntu waawsanee bialssalati waalzzakati ma dumtu hayyan

19:31 (Y. Ali) "And He hath made me blessed wheresoever I be, and hath enjoined on me Prayer and Charity as long as I live;

because sleep = die.

Az-Zumar (The Groups)

Allahu yatawaffa alanfusa heena mawtiha waallatee lam tamut fee manamiha fayumsiku allatee qada AAalayha almawta wayursilu alokhra ila ajalin musamman inna fee thalika laayatin liqawmin yatafakkaroona

39:42 (Y. Ali) It is Allah that takes the souls (of men) at death; and those that die not (He takes) during their sleep: those on whom He has passed the decree of death, He keeps back (from returning to life), but the rest He sends (to their bodies) for a term appointed verily in this are Signs for those who reflect.

Salam.

Therefore there is clear mention in the Quran for 'salat' (prayer) to be enjoined / prescribed on believers and at specific times / periods of the day.

In light of 17.78, the prayer is establised continously.

Al-Isra (The Journey by Night)

Aqimi alssalata lidulooki alshshamsi ila ghasaqi allayli waqurana alfajri inna qurana alfajri kana mashhoodan

17:78 (Y. Ali) Establish regular prayers - at the sun's decline till the darkness of the night, and the morning prayer and reading: for the prayer and reading in the morning carry their testimony.

Salam.

[4:103]

With regards to 'salata' (prayer), verse 4:103 makes use of the Arabic 'kitaban mawquta' which refers to 'prescribed' times.

I think 'prescribed' = kutiba,

and

Although 'waqt' from which 'mawqutun' is formed means a state, fixed time or a period of time,  the word 'kitaban' means prescribed / enjoined / appointed / inscribed what is written or ordained.

'kitaban' = prescription