Simple circuit help?

Assume that such a circuit exists . When you connect the cable to the 

circuit a chattering may occur at the connecting points so LED lights 

and turns off , never lights again until you push the reset button . 

So I slightly change your specification . 

Use one SR flip flop . 

S'. R' Q. Q' (S'=inverted set input , R'=inverted reset input) 

H  H  ....... (holds former status) 

L . H  H . L 

H  L . L . H 

L . L . H. H (note) 

note. If S' and R' go to H simultaneously then the flip flop may go to 

an unstable status . But if S' and R' never go to H simultaneously then 

this status is not inhibited . 

LED lights when Q' = L . 

S' is pulled up with proper resistor , and a push-on "Start" button switch 

is connected between S' and GND . So usually S' = H and it becomes L

when "Start" button is pushed . 

An inverter is connected to R' , its input is R and R is pulled up with 

proper resistor . The cable is connected between R and GND , so usually 

R = L (R' = H) , it becomes R = H (R' = L) when the cable loses its 

conductivity .  

This circuit operates as followings . 

Step1. Turn on the system power without connecting the cable . 

R' = L and S' = H so Q' = H and LED is turned off . This step settles the 

status of the flip flop . 

Step2. Connect the cable between R and GND . 

If the conductivity exists then R becomes L (R' becomes H) , so Q' is 

still H and LED is still turned off . 

Step3. Push "Start" button . 

If the conductivity exists then R' = H and S' becomes L so Q' becomes L 

and LED lights . 

If the conductivity is lost then R' = L , so when S' becomes L , Q' is still H

and LED does not light . You must check the connections , and if the 

connections is proper then remove this cable and go to step2 with a new 

cable .  

Step4. Test the cable . 

You can stress the cable out . 

If the conductivity is lost then R becomes H (R' becomes L) , so 

Q' becomes H and LED turns off . If the conductivity is restored then 

R becomes L (R' becomes H) but Q'is still H and LED is still turned off . 

***** 

When you must change the threshold level of the conductivity , a voltage 

comparetor can be used .

Your body is a good earthing for very small voltages. Take LED of a given value. Now connect a resistance parallel to LED and adjust the resistance such that led glows after completing the circuit. Now you have two end one from LED and one from power power unit. Connect the two you get  LED glow. You can check the continuity  with it. After that just touch with finger to the two ends of led. This will work. If you are using it with main power then you need a set setup similar to testers.

I couldn't be bothered designing it for but you'd definitely need a flip flop for that.  And yes, you most certainly don't need to use an Arduino for that.

Update:  I'm talking about a latch, like an RS flip flop, D flip flop, JK flip flop etc.  Whatever you think is best.  Anyway, what's a resistor flip flop?

Classically, you would have used a flipflop for that. The ubiquitous 555 in bistable mode should be able to do it - possibly with an inverter on the wire, I haven't drawn up the full circuit.

In practice, though - well, I work for an electronics manufacturer. For our test setups, we needed a thingy that would create a defined reset pulse when a button was pressed. Classical 555 territory, you might say. However, it turned out that, for little more than the price of a bare 555 chip, we could get a complete arduino nano module - complete with push button, PCB and a USB connector.

If you want to go back to the basics, though - what you're asking is also known as bi-stable multivibrator. Wikipedia has a circuit diagram on it: https://en.wikipedia.org/wiki/Multivibrator#Bistab... . You'll need to add the LED(s) and possibly driver transistors yourself.