using one database connection across multiple functions in python

The best thing it occurs to me it's creating a class, so you can save the connection in an attribute, and then use it by all the test methods. In example:

import sys
import MySQLdb as mdb

class MyDBTest():

    def __init__(self, host, user, passw, db):
        try:
            self.con = mdb.connect(host, user, passw, db)
            self.cur = self.con.cursor()
        except mdb.Error, e:
            print "Error %d: %s" % (e.args[0], e.args[1])
            sys.exit(1)

    def show_version(self):
            self.cur.execute("SELECT VERSION()")
            print "Database version : %s " % self.cur.fetchone()

    def test1(self):
        # Do something with self.con or self.cur
        pass

    def test2(self):
        # Do something with self.con or self.cur
        pass

if __name__ == '__main__':  # If it's executed like a script (not imported)
    db = MyDBTest('10.243.17.13', 'mark', 'LiverpoolFc', 'dex')
    db.show_version()
    db.test1()
    db.test2()

In the example I've put the error checking (try-except) inside the class. But you could also don't put it in there and leave the main (__name__ == __main__ part) to handle the errors if they occurr inside MyDBTest.