Take the meter, put it on the low ohms range and connect the probes together, this should give zero ohms/short circuit. You may need to adjust the “ohms” wheel on the left to get zero. Remember that the meter should be on the infinity mark before you short the leads; adjust with the screw in the centre of the meter movement to get the meter zeroed properly.
Connect the meter to the inner and outer and you should see no deflection on the meter. Some antenna types will show a DC path but a discone should not.
Assuming no deflection, disconnect the antenna and connect a wire across the inner and outer at the antenna end…you should now see a dead short; zero ohms, or thereabouts.
At this point that is pretty much as good as you can do with testing coax with a multimeter; here you should have shown no DC path between the inner and outer but also continuity through the connector outer, into the screen, up the screen, through the top connector back into the centre conductor and back down again.