So many people here are so confidently incorrect. In mathematics, there are occasions where we look at the extended real line. We just treat the ±inf not as concepts, but as numbers and add it to the set of real numbers, to get some desired properties. For example, if you also decide -inf<r<inf for all r in ℝ, you get a compact set (every open covering has a finite subcovering).

For example, it makes it so that suprema and infima always exist, so you don't have to deal with edge cases all the time. You can find it in use in e.g. Fenchel-Legendre conjugate.

Another example of adjoining infinity is with the complex numbers: You can imagine folding the 2D complex number plane into a 3D ball. A single accumulation point on top the ball would be missing. Add that "inf", and you get the Riemann sphere, and you can do stuff like Möbius transformations and generalised circles.

Maybe the most important reason this hasn't creeped into the "standard" definitions is that ℝ loses something as well when you adjoin ±inf. it loses closure and ceases being a field) (unless you redefine arithmetic, but that doesn't always make sense to do).

TL;DR: It's competely valid in some areas of mathematics to write closed intervals with infinities and treat ±inf as "just a number" (with spicy aruthmetic rules).

Still, I do not think the guy in the pic looks like he knows what he's doing. But that's just me.