When something is threadbare or worn, you can describe it as frayed. Your hand-knit mittens will probably become frayed around the cuffs after many winters.
If your favorite pair of jeans is frayed, it may be because you've worn them for years — or they could be deliberately, stylishly frayed from the minute you buy them. Things that are frayed are worn, even unraveling a bit. The earliest meaning, in fact, was "worn by rubbing," from the Middle French frayer, "rub against," by way of the Old French froiier, "rub or scrape," and the Latin root fricare, "to rub."