How to turn a pulsed signal into a continuous one?
You are failing to see the error in your belief. 

If the pulse is high (as normal) and then goes low (as normal) and while low it is "interrupted", the signal will still be low and you won't have known the signal was interrupted until you realize that it has failed to go back high some time later.

It's like when your phone rings - it does so in bursts - you go to pick it up when the bell falls silence but you don't actually know that the person ringing you has hung up during that silence.

So forget about "immediately" and use a re-triggerable monostable. Here's the waveforms: -



Here's a retriggerable monostable circuit: -



Picture stolen from here. Basically, a re-occuring pulse fed to the input (the little circle to the left in front of the two resistors feeding a BJT) keeps the 555 from timing out.