
haften (für)
to be liable for
To be legally or financially responsible for damage, a debt, or an obligation that has occurred, and to have to answer for it.
Eltern haften für ihre Kinder.
Parents are liable for their children.
to be liable / to stick
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to be liable for
To be legally or financially responsible for damage, a debt, or an obligation that has occurred, and to have to answer for it.
Eltern haften für ihre Kinder.
Parents are liable for their children.

to stick to
To stick to a surface or adhere firmly to it.
Der Aufkleber haftet sehr gut auf der glatten Oberfläche.
The sticker sticks very well to the smooth surface.

to remain
To remain in memory or recollection.
Sein seltsames Lächeln ist mir im Gedächtnis haften geblieben.
His strange smile has remained in my memory.
The preposition after 'haften' changes the meaning completely. 'Haften für' means to take responsibility, while 'haften an/auf' describes physical sticking.
Der Aufkleber haftet nicht gut auf der rauen Wand.
The sticker does not stick well to the rough wall.
In the legal meaning, 'haften' always appears with the preposition 'für' followed by the accusative. The structure is: someone is liable for something/someone.
Laut Vertrag haftet die Firma für alle Lieferverzögerungen.
According to the contract, the company is liable for all delivery delays.
When 'haften' means 'kleben' (to stick), it is used with the prepositions 'an' or 'auf' and the dative. It describes where something is physically attached.
Nach dem Spaziergang im Wald haftete Schmutz an meinen Schuhen.
After the walk in the forest, dirt was sticking to my shoes.
The expression 'im Gedächtnis haften bleiben' (to remain in memory) is a fixed combination. It describes a memory that is very strong and present. The person is in the dative.
Diese besondere Melodie ist mir seit meiner Kindheit im Gedächtnis haften geblieben.
This special melody has remained in my memory since my childhood.
to hang
Hängen and haften can both describe something being attached to a wall. With hängen, the object is supported by a single point or a holder and hangs down. With haften, it stays attached through direct contact with the surface.

Das Schild haftet mit Klebestreifen direkt an der Wand.
The sign sticks directly to the wall with adhesive strips.
Das Schild hängt an einem Haken an der Wand.
The sign hangs from a hook on the wall.