You still need to worry about the domain, in general there are plenty of real valued discontinuous linear functions (or functionals) from an infinite dimensional Banach space to R.

In finite dimension however linear functions are continuous

That is a badly written article. And the ones for linear map and continuity linked there, which would be important to the topic, too. The good mathematical articles start with a definition after the intro.

So much for teaching year olds. This is not rocket science, but mathematicians (almost inherent) inability to have language skills makes it seem so, notably.