Finding exact breakpoint location in long wire
This is accomplished with a time-domain reflectometer (TDR).

You know that in a transmission line, an impedance mismatch will cause a voltage reflection. Then you just need to know the propagation speed of voltage along the transmission line. At that point you can transmit a test pulse and time how long it takes for the reflected voltage to appear. Propagation speed multiplied by time equals the distance travelled. Because the wave had to go there and back, half that calculated distance is the length of cable to the break.

You can think of this as echolocation for transmission lines.

Edit:

In practice, the propagation speed of the voltage is usually between 42% and 99% the speed of light. This is also called the velocity factor.


  The VF of a lossless transmission line is given by:
  
  $$VF = { \frac{1}{\sqrt{LC}} } \ $$ Where L is the distributed
  inductance (in henries per unit length) and C is the capacitance
  between the two conductors (in farads per unit length).