Stupid question about eyeshadow liner pencil

[noblegiraffe]

I was given an Astor 24 hour perfect stay eyeshadow and liner pencil, which was great. The pencil wore down a bit and I have no idea to get it sharp again/get more pencil. The case appears to be plastic and doesn't sharpen like a normal pencil and a bit on the bottom that looks like it might twist to push up more pencil is immovable. Surely the tiny sharp point at the top wasn't all there was to it?

For the lack of anyone else, at the moment, I will reply :) DD had something like this, that looked like it wouldn't sharpen, but it did. Have you actually tried to sharpen it? Can't quite decide from your OP. Ignore if you have!

Have just Googled and they do sharpen