durchgängig
continuous
Without interruption; continuously from beginning to end. Can refer to time or a process.
Das Museum ist von 10 bis 18 Uhr durchgängig geöffnet.
The museum is open continuously from 10 a.m. to 6 p.m.

continuous / passable
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continuous
Without interruption; continuously from beginning to end. Can refer to time or a process.
Das Museum ist von 10 bis 18 Uhr durchgängig geöffnet.
The museum is open continuously from 10 a.m. to 6 p.m.

The word consists of 'durch' (through) and 'gängig', which comes from 'gehen' (to go). The basic idea is that something can 'durchgehen' (go through). This fits all meanings: a period of time without a pause, a rule without exception, or a path that is passable.
Der Radweg ist von der Stadt bis zum See durchgängig befahrbar.
The bike path is continuously rideable from the city to the lake.
'Durchgängig' (continuous/consistent/passable) can be used as an adjective before a noun or as an adverb referring to the verb. Its position in the sentence changes its function, but not the basic meaning.
Wir brauchen ein durchgängiges Farbkonzept für die ganze Wohnung.
We need a consistent color concept for the whole apartment.
The context makes the meaning clear. If it is about time, such as opening hours, the meaning is 'ohne Unterbrechung' (without interruption). If it is about rules or concepts, it means 'ohne Ausnahme' (without exception). If it is about paths or roads, it means 'passierbar' (passable).
Die Software wurde durchgängig in englischer Sprache dokumentiert.
The software was documented consistently in English.
In the time-related meaning, 'durchgängig' (continuous/continuously) is often combined with verbs such as 'geöffnet sein' (to be open), 'erreichbar sein' (to be reachable/available), or 'besetzt sein' (to be staffed), usually with reference to services.
Unser Kundenservice ist von 8 bis 20 Uhr durchgängig erreichbar.
Our customer service is continuously available from 8 a.m. to 8 p.m.
consistent
'konsequent' and 'durchgängig' can both express that there is no exception. 'konsequent' is used especially for people, decisions, or actions that stay true to a consistent line. 'durchgängig' tends to describe something that is the same from beginning to end or in all parts. It can refer to time, a text, a plan, or a system. There is a risk of confusion because both can mean 'without a break'. As a guide: people act 'konsequent'; a pattern, a rule, or a quality is 'durchgängig'.

Der Bericht ist durchgängig klar geschrieben, sodass alle ihn gut verstehen.
The report is written clearly throughout, so everyone can understand it well.
Die Eltern bleiben konsequent, obwohl das Kind lange diskutiert.
The parents remain consistent, even though the child argues for a long time.