What's an effective way to remove this stubborn burnt-on oil from a saucepan? I have a stainless steel saucepan that has dark burnt on oil around the sides. I'm assuming it's from cooking oil since I use oil in this saucepan a lot.

It seems to just keep building up and up and I can't get rid of it.

This morning I tried boiling 1cup water with 1cup distilled vinegar for 5 minutes, then adding baking soda. I emptied the water, then tried to scour it clean but it made very little difference.

Is there some industrial strength stuff I can buy to help me out here?

Here's a pic: https://imgur.com/a/G2CkM
If your pan is truly stainless steel, you can get burned on brown and sticky oil off the sides with oven cleaner (US brand "EZ-off" or other brand). This product is definitely not recommended for other metals such as copper, aluminum, etc.. But true stainless steel, I have used it no prob. If you are in doubt, you can test a small amount out on an inconspicuous tiny spot at first.