[gunnihinn]

A distribution is not a function. It is a continuous linear functional on a space of functions.

Functions define distributions, but not all distributions are defined that way, like the Dirac delta or integration over a subset.

[skhunted]

A functional is a function.

[ogogmad]

The term "function" sadly means different things in different contexts. I feel like this whole thread is evidence of a need for reform in maths education from calculus up. I wouldn't be surprised if you understood all of this, but I'm worried about students encountering this for the first time.