this defines my favorite function 

Definetly

Nooo! You spoiled me the video.

I guess I'm a real fan and I just didn't know it, because I check every day for uploads but I didn't recognize the inequality as specifying the floor function.

why is this the top comment, fucking spoilers

​@@9WEAVER9It's not about the inequality, but rather fhe video's title.

I think I was able to guess from the statement alone. The inequality makes it look like the function is sort of linear (with an error of about 1 unit at most), and it also specified that f(x)

Yesa, I like them, too. But I am not good at solving most of the time.

​@@hassanalihusseini1717 they are fun

I asked myself the same questions and have some partial answers. There are other solutions then. If, for instance, you drop the red condition, you can multiply by any element a

try it yourself

y=x also works, right?

​@@lyrimetacurl0not quite... There's the not-in-thumbnail inequality 2

Yes.

technically he only proved that if a function with these properties exist, then it must be the floor function. however it is rather easy to see that the floor function indeed satisfies the stated conditions

I thought it was wrong at first, too. I had to analyze it a bit to understand. Lol I think swapping "f(x) + f(y) + 1" and "f(x) + f(y)" and flipping the inequalities would make it less confusing.

It's not

​@@assassin01620 it is saying that f(x)+f(y)+1

@@lucianoxiccato5458 The red arrows are telling you where to place the "f(x) + f(y) + 1" and the "f(x) + f(y)". So it's saying: "f(x) + f(y)

@@assassin01620 AHHHHH sorry but that was confusiong ahah

you're missing the red arrows

@@MrStanny32 indeed I did. I think reversing the three lines would be clearer still.

Just a variation of the & symbol, used within linguistics too in some cases. Which one to use is usually just up to the preference of whoever's writing it.

As is said below, that's just a graphical variant of the ampersand symbol. The ampersand symbol is just a digraph (read: combination that retains the original order) of "et", the Latin word for "and". That variant symbol is really supposed to be an E with the two lines above and below representing the t, but, for some unknown reason Michael wrote the symbol backward. (Yes, they aren't really dots, but small line segments, kind of like how some people don't put a full line through the S when writing a $, but just put little lines on the top and bottom of the symbol.). There's another variant that's sometimes used that is literally just a curvy Et, with the bottoms joined together (It helps if you use the lowercase t with the serif on the bottom, like in the font RU-vid uses, but flipped the other way around.). If you write that and kind of tilt the top of the T leftward a little, then you'll get an ampersand. Also, I'd like to point out that the name "ampersand" comes from two facts: one is that the ampersand used to be considered an actual letter of the alphabet in English, and the second is the Alphabet Song. In the song, the last letter was the ampersand, but it was originally called "per se and". "Per se" is Latin for "for itself", so "per se and" meant the symbol was the word "and" in its own right. Since "per se and" was the last letter of the alphabet, the song ended with the words "and per se and", which eventually was re-bracketed and shortened to "ampersand".

It doesn't. But if you know the values from 0 to 1 (which he calculated earlier) you can use this formula to calculate all of the others.

That is wring.every function is special in themselves.

The inequality immediately tell you that atleast the floor function is one of the solutions