
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.

consistent
Applicable to all parts or cases; consistent and without exception.
Die neue Regelung wird durchgängig auf alle Abteilungen angewendet.
The new regulation is applied consistently to all departments.

passable
Designed or in a condition so that one can walk or drive through it; passable.
Nach dem Unfall war die Straße für Stunden nicht durchgängig.
After the accident, the road was not passable for hours.
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.
continuous
Both words can refer to a process without interruptions. „Kontinuierlich“ mainly describes its steady development over time, while „durchgängig“ emphasizes that a property applies to all parts or phases. Use „kontinuierlich“ for the development and „durchgängig“ for a consistent property.

Der Kurs war durchgängig anspruchsvoll, von der ersten bis zur letzten Aufgabe.
The course was challenging throughout, from the first task to the last.
Der Kurs wurde kontinuierlich schwieriger, weil jede Woche neue Aufgaben dazukamen.
The course became continuously more difficult because new tasks were added every week.