festhalten
to hold
To grip something or someone firmly with the hands or arms and not let go.
Bitte halte das Bild gerade, während ich den Nagel in die Wand schlage.
Please hold the picture straight while I hammer the nail into the wall.

to hold / record
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to hold
To grip something or someone firmly with the hands or arms and not let go.
Bitte halte das Bild gerade, während ich den Nagel in die Wand schlage.
Please hold the picture straight while I hammer the nail into the wall.

The verb is made up of 'fest' (firmly) and 'halten' (to hold). The literal meaning 'to hold something firmly in one's hand' is the basis for all figurative meanings, such as holding on to an idea or recording information.
Er hielt den entscheidenden Moment in einem Foto fest.
He captured the decisive moment in a photo.
'Festhalten' (to hold / record) is a separable verb. In a main clause, the prefix 'fest' goes at the end. In subordinate clauses and in the infinitive with 'zu', the verb stays together.
Halte bitte meine Tasche kurz fest.
Please hold my bag for a moment.
In the literal meaning, the reflexive form 'sich festhalten' (to hold on) is often used, usually followed by the preposition 'an' with the dative. This describes giving oneself support.
Das Kind hielt sich ängstlich an der Hand seiner Mutter fest.
The child anxiously held on to his mother's hand.
In the sense of 'dokumentieren' (to document), 'festhalten' (to record) is combined with abstract nouns such as 'Ergebnisse' (results), 'Ideen' (ideas), 'Gedanken' (thoughts), or 'Punkte' (points). The place or form of the record is often mentioned, e.g. 'im Protokoll' (in the minutes) or 'schriftlich' (in writing).
Wir sollten diese Idee schriftlich festhalten, bevor wir sie vergessen.
We should record this idea in writing before we forget it.
To express that someone is sticking to an opinion or plan, 'festhalten' (to stick to) is always used with the preposition 'an' and the dative. The pattern is: an etwas (dative) festhalten.
Sie hält an ihrem ursprünglichen Plan fest, obwohl es Schwierigkeiten gibt.
She is sticking to her original plan, even though there are difficulties.
The meaning 'to prevent someone from leaving' is used almost only in the context of police, security, or crime. In this case, the object is always a person.
Der Ladendetektiv hielt den Dieb bis zum Eintreffen der Polizei fest.
The store detective detained the thief until the police arrived.
To emphasize an important statement, 'festhalten' (to state) is often used in fixed formulations such as 'Ich möchte festhalten, dass ...' (I would like to state that ...) or 'Es lässt sich festhalten, dass ...' (It can be stated that ...). This introduces a conclusion or a central point.
Lassen Sie uns festhalten: Das Projekt wird wie geplant fortgesetzt.
Let us state this clearly: the project will continue as planned.
to stay / remain
'bleiben bei' and 'an etwas festhalten' can both mean that someone does not change an opinion, decision, or plan. 'Bleiben bei' often sounds neutral: you have made a decision and do not change it. 'An etwas festhalten' sounds stronger. It is especially suitable when there is pressure, doubt, or resistance, but someone still does not give up. The words can be confused because both express consistency. As a simple guide: 'Ich bleibe bei meiner Entscheidung' (I am sticking with my decision) is neutral, while 'ich halte an meiner Entscheidung fest' (I am holding firmly to my decision) emphasizes firm determination.

Auch nach der Kritik hält Lisa an ihrer Meinung fest.
Even after the criticism, Lisa is sticking to her opinion.
Trotz der neuen Informationen bleibt der Chef bei seinem Plan.
Despite the new information, the boss is sticking with his plan.