
durchgehend
continuous
In terms of time, without interruption; uninterrupted.
Die Rezeption des Hotels ist durchgehend besetzt.
The hotel reception is staffed continuously.
continuous / throughout
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continuous
In terms of time, without interruption; uninterrupted.
Die Rezeption des Hotels ist durchgehend besetzt.
The hotel reception is staffed continuously.

direct / continuous
Extending spatially from one end to the other; without changing or transferring.
Es gibt eine durchgehende Zugverbindung von Berlin nach Prag.
There is a direct train connection from Berlin to Prague.

consistently
Used as an adverb: generally, without exception, across the board.
Die Kritiken für den Film waren durchgehend positiv.
The reviews for the film were consistently positive.
The word is made up of 'durch' (through) and 'gehend' (present participle of 'gehen' (to go)). The literal meaning 'going through' helps you understand the core idea of uninterrupted continuation in time or space.
Eine durchgehende rote Linie markiert den Radweg.
A continuous red line marks the bike path.
'Durchgehend' (continuous/throughout) can stand as an adjective before a noun or as an adverb describing a verb or a whole sentence. As an adverb, it often stands after the verb or at the beginning of the sentence for emphasis.
Wir haben eine durchgehende Arbeitszeit von 9 bis 17 Uhr.
We have continuous working hours from 9 a.m. to 5 p.m.
In its time-related meaning, 'durchgehend' (continuous/continuously) is often combined with services or states. Typical phrases are 'durchgehend geöffnet' (open continuously), 'durchgehend besetzt' (staffed continuously), or 'durchgehend erreichbar' (available continuously).
Unser Kundenservice ist telefonisch durchgehend erreichbar.
Our customer service is continuously reachable by phone.
In its spatial meaning, 'durchgehend' (direct/continuous) often describes transport connections. Common combinations are 'durchgehender Zug' (direct train), 'durchgehende Flugverbindung' (direct flight connection), or 'durchgehende Straße' (continuous road).
Gibt es einen durchgehenden Flug nach Lissabon oder muss ich umsteigen?
Is there a direct flight to Lisbon, or do I have to change planes?
In the meaning 'ausnahmslos' (without exception), 'durchgehend' (consistently) is used as an adverb to describe a general tendency. It then often modifies an adjective or describes the quality of an action.
Die Schüler haben in der Prüfung durchgehend gute Ergebnisse erzielt.
The students achieved consistently good results in the exam.
direct
When talking about travel, direkt and durchgehend can describe a connection without transfers. Direkt generally refers to a connection without changing trains, while durchgehend emphasizes that it continues from the starting point all the way to the destination. Both connections may still make stops along the way.

Dieser Zug fährt durchgehend von Köln nach Basel, sodass wir nicht umsteigen müssen.
This train runs all the way from Cologne to Basel, so we don’t have to change trains.
Wir nehmen die direkte Zugverbindung von Köln nach Basel und müssen nicht umsteigen.
We’re taking the direct train from Cologne to Basel, so we don’t have to change trains.