Replacing the wiring to a socket in a home?
To confirm the wire is broken/damaged, you can measure the electrical resistance of the cable with a multi-meter.  If the wire is OK, you should get a near-zero measurement.  If it is broken or significantly damaged, the resistance will be a high value or infinity.

You need to isolate the section of cable you think is damaged to in between two electrical boxes.  With the power off, at one end of the cable, connect the hot and neutral with a wirenut.  Measure the resistance between the two at the other end. Repeat this with all combinations of hot/neutral, hot/ground, neutral/ground in order to figure out which wire is broken.

As far as replacing the cable, you will likely need to cut one or more holes in your wall in order to pull a new cable through.  If it's a relatively straight and short run you might get away just removing the electrical boxes and fishing a new cable through, and then replacing the electrical boxes, but usually you have to cut multiple holes to run a new cable.