Q: Lightning cable Outer rubber protection is breaking...it is under warranty? Will they replace it?

The cable caries a 12 month warranty just like all other Apple products. If the problem is caused by a manufacturing defect, it will be replaced. If it is caused by user mishandling, it may not.  Your damage looks typical of pushing and pulling on the cable rather than the connector.

Take it to Apple. It's their decision, and you might get lucky.

I keep having the same issue, but mine is in a worse condition to yours. No matter how gentle I am, it still seems to do that. I've been through about 3 cables since I got my 5C just over a year ago. Apple replaced the cable the first time and then my network provider replaced it last time. Now my network provider wont replace it, they ordered me a plug replacement by mistake. Hopefully Apple will, but don't have an Apple store close to where I live. My current cable is about 2 or 3 months old.

I have the same issue. My warranty is over long time ago. Condition is about the same as on the picture.

Is it safe to use it? Or should I through that away?

Thank you.