behaupten
to claim
To state something as true or accurate, often without providing proof for it.
Er behauptet, die Brieftasche auf der Straße gefunden zu haben.
He claims to have found the wallet on the street.

to claim / assert
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to claim
To state something as true or accurate, often without providing proof for it.
Er behauptet, die Brieftasche auf der Straße gefunden zu haben.
He claims to have found the wallet on the street.

'Behaupten' (to claim/assert) is an inseparable verb because it begins with 'be-'. The prefix remains firmly attached to the verb in all sentence forms and tenses.
Sie behauptete, von nichts gewusst zu haben.
She claimed to have known nothing about it.
When making a claim, a subordinate clause with 'dass' (that) or an infinitive construction with 'zu' (to) often follows. The structure is: jemand behauptet, dass ... / jemand behauptet, ... zu tun (someone claims that ... / someone claims to do something).
Der Zeuge behauptet, dass er den Täter gesehen hat.
The witness claims that he saw the perpetrator.
The meaning changes completely when the reflexive pronoun 'sich' is present. Without 'sich', it means 'etwas aussagen' (to state something) (m01); with 'sich', it means 'sich durchsetzen' (to assert oneself / prevail) (m02).
Er behauptet, der Beste zu sein.
He claims to be the best.
The word 'behaupten' (to claim) often implies that the speaker distrusts the statement or that there is no evidence for the statement. It expresses a certain distance or skepticism.
Er behauptet, ein berühmter Schauspieler zu sein, aber niemand kennt ihn.
He claims to be a famous actor, but no one knows him.
The reflexive form 'sich behaupten' (to hold one's own) is often used in the context of competition and rivalry. Typical prepositions are 'gegen' (against) for competitors or 'am' (in/on) for the market.
Das neue Produkt konnte sich schnell am Markt behaupten.
The new product was quickly able to hold its own in the market.
to say
'sagen' and 'behaupten' can both mean that someone makes a statement. 'sagen' is neutral and very general. You can say a piece of information, an opinion, or advice. 'behaupten' is used when someone presents something as true, but you are not sure whether it is correct, or when proof is missing. That is why 'behaupten' often sounds more cautious or more critical than 'sagen'. Learners may confuse the words because both describe making a statement. If you only want to report something neutrally, use 'sagen'. If there is a sense of doubt about the statement, 'behaupten' fits better.

Der Schüler behauptet, dass er die Hausaufgaben schon abgegeben hat.
The student claims that he has already handed in the homework.
Im Park hat er gesagt, dass das Treffen morgen stattfindet.
In the park, he said that the meeting is taking place tomorrow.