>We've been working on a vertical app.
>
>We're really close on being finished with it. Here's the thing. We saw a demo of a product offered by a competitor.
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>There's a thing that our application has that our competitors does'nt fully offer.
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>But that particular competitor (the demo we saw) says in his publicity that their application is offering it. and in fact it's something that is not really completed yet.
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>
>So from that publicity here's what I understand.
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>. That competitor realized that this characteristic could become very important in the future.
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>. They probably intend to make their software support what they say as fast as they can but they're not there so far so they just fake it in their publicity.
>
>
>How should we react to their (could we say false) publicity?

No you have not right to say this.
They have right to adverize or present their software in a way which they think is the best. If some user catch them in false they have right to bring a suit against them.
They can set a law suit if you say public that this is false. I suppose that they hardly work to add this feature.

You have to be happy that you put them in situation to add new features - non-stop changes lead to software instability. Which could be advantage for you