Would Apple replace a worn out charging cable?

Hello,

I purchased an iphone5 last year in May and within 8 months of purchase, its charging cable insulation started getting loose near its end and now it has worn out. Yesterday the charging cable developed an inch long cut more on it even the already worn out part was covered with an insulated tape.

I went to the apple store last month and they said they won't replace it coz its still in the working condition.
Everytime I look at the charger, it makes me feel so bad..Huh..

Why has been the compromise made with the quality of insulation? Has this been my fault, I would have agreed to it but tearing/wearing out of its own within a year of warranty and still not replacing it makes me so disappointed.

I want to know if the charging cable would be replaced, its still in the warranty period. I would appreciate and would be so happy if Apple replaces the damaged iphone charging cable. 😟

Regards

Samriti