Yes and no. As soon as the user moves to the next page, for all purposes and tasks they are no longer connected to the database.
Your script (on the next page) will still need to open a connection for them. mysql_pconnect() will ensure that the actual connection they used is still available when it wants to, but it can also cause an excessive number of apache/mysql connections to wait uselessly.
I would strongly recommend not using it unless your tests show that it provides a significant increase in performance. As a rule, for most applications (especially when you are studying), I will not worry about constant connections. Note the warning in the PHP manual
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