Just like in the question, be able to determine when the player has begun typing within the chat box and know when they stopped typing / close the chat
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How would I determine when a player is currently typing in the chat box?
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✓ Accepted answer
In the text box, use :IsFocused() to see if a player is typing in the chatbox. http://wiki.roblox.com/index.php?title=API:Class/TextBox is where you will find all the 'focus' items.
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With the UserInputService's InputBegan, it comes with a second argument that checks if a player's typing or not. (Arg1 = Key pressed, Arg2 = Typing (use
if not Arg2b/c that's when they're typing))