A person who goes by his own pace regardless how hectic or unnatural things are going on in his surroundings. General way of distinguishing them is noticing their calm and rational behaviour as if things are in their control (even if it isn't then it doesn't matter to them as they will still do what they were previously doing).
1. There is a terrorist attack in your city and you still thinking of if they will be airing Aldnoah:Zero on TV or not? As expected of a my pace person, nothing fazes you.
2. (Among friends, topic of discussion is regarding politics or anything that isn't related to food)
Otsu : Using this policy could be certain setback to our economy in the country. What do you think about it Nao?
Nao : I wonder if eggs are going to sale faster tonight (setting up the reminder in phone to buy eggs early).
Otsu : Oh my dear god Nao, we are discussing about how economy could be affecting and you still bothering if you will get to eat eggs or not? You are being my pace again.
(Economy will result change in price shift for good or not. So actually the food sale comes as it's after effects which could make people think of hoarding goods. This example shows Nao's calm and rational thinking to public outcry which could affect him while being unfazed while the discussion was still on preliminary stages and yet to reach that point)
3. The gif is from anime Aldnoah:Zero. The friends still talking about the visiting princess while the main character calmly notices the launched missile and warns his friends. Suddenly breaking news of missile so calmly they don't react to it until the bomb blasts happen nearby.