Do you have a volt-ohmmeter (VOM)? If so, check the continuity from the center pin to the shell of the connector. Should be an open circuit; no continuity. Then shell to the mount bracket, should have continuity. Then center pin to where the Firestik screws in, should have continuity.
If all that checks out, pull the end cap off the Firestik and check for continuity from the part that screws in to the end of the wire under the cap. You might have to scrape the wire with a knife or sandpaper to make sure you get a good connection to the meter. That should have continuity too.
Occasionally that wire doesn't make a good connection at the bottom of the antenna.
You don't necessary need a VOM to check these things. A battery and test light works too. It's also pretty simple to make a test light out of a cheap flashlight.