Induction on its own does not mean continuous monitoring.
I have been induced five times, it depends on the method of induction and how it is going as to if you need continuous monitoring.
If you end up needing the syntocinon drip they like to use continuous monitoring as the contractions can be stronger and baby may become distressed. If you just have the pessary or your waters broken and thst gets labour goung then you don't need continuous monitoring. I have had pool birth even with being induced and in the labour where I did have the drip and hence continuous monitoring I still stayed mobile, active using birth ball, walking about etc. It was more awkward due to wires but the midwife helped with this. And as others have said some hospitals have the wireless monitors now.
There is evidence that continuous monitoring can lead to more intervention but doesn't improve outcomes.
So it's down to choice, I have always opted for intermittent monitoring and taken advice of midwife. It's a case of weighing up risks, V benefits as with everything in pregnancy and childbirth there is no zero risk option so you have to do what you are comfortable with.
I need to move in labour and use water for pain relief if I can so intermittent monitoring with hand held doppler was best for me. The newer wireless monitors sound great, I think my hospital has these now so if I end up induced for no 6 due April and for any reason continuous monitoring is necessary I will be asking for the wireless monitoring but I would rather just intermittent monitoring.
It really is something that depends on individual circumstances and your preferences.