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The phrase "if invited" is correct and usable in written English.
It can be used to indicate a condition where something will happen only in the case that an invitation is extended. Example: "I will attend the party if invited." Alternatives: "should I be invited" or "if I receive an invitation."
Exact(55)
They are ready to go if invited.
Two artists announced that they would not participate if invited.
Bahrain said it would send naval forces if invited.
Where their neighbors may come only if invited.
But she said she felt good Sunday and would return to Chicago next year, if invited.
Bogdan was never to enter this room, not even if invited.
Calhoun was noncommittal on playing in the N.I.T., even if invited, given this team's lackluster play.
Moser asks what Forman would do if invited to be director general for the day.
But British workers are permitted to indulge modestly if invited by their French counterparts.
You will be asked about what you can contribute to the school if invited for interview.
He expanded it to schools after lawmakers gave him permission to scrutinize local school districts if invited.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com