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.

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 wear / carry
'tragen' and 'haften' can both express that someone has to bear consequences or costs. 'tragen' is more general: you can bear costs, responsibility, or consequences. 'haften' is used especially in legal or financial situations. It means that someone is responsible for damage, a debt, or an obligation and has to pay or answer for it. The words can be confused because both have to do with responsibility. However, 'haften' is narrower and more legal; 'tragen' also fits in normal everyday situations.

Bei einem Schaden haftet der Mieter für die Reparatur.
If there is any damage, the tenant is liable for the repair.
Die Eltern tragen die Kosten für die Klassenfahrt.
The parents bear the costs of the class trip.