What is the function $E(x)$?

Question

When reading Problems in Calculus of One Variable (a translated Russian book), I came across unfamiliar notation "$E(x)$". It is neither expected value nor $\exp(x)$. Here is a picture of the function used in context, which I hope someone can deduce what it means from

$$\int\limits_0^x E(x)\mathrm d x=\frac{E(x)(E(x)-1)}{2}+E(x)[x-E(x)]$$

It is not defined in the book, nor specific to context, and also used in multiple instances.

I don't think anyone can guess which function the authors had in mind (though maybe someone could reverse engineer $E(x)$ from that integral). I expect it is defined somewhere in the text. — lulu, Jul 23 '20 at 12:04
I don't know what it is but $\int_0^{x} E(x)dx$ is not an acceptable notation for any function $E(x)$. — Kavi Rama Murthy, Jul 23 '20 at 12:06
Thomas' Calculus used that notation to denote an arbitrary even function. Not sure about this one, though. — Manan, Jul 23 '20 at 12:06
@lulu As said in the post, it is not defined in the book. I find it unlikely the author used his own notation and did not define it only for readers to guess at, so I assume someone will know it, possibly from a different part of the world, unlike us. — twentyyears, Jul 23 '20 at 12:06
See 1.14.4: "the function $E(x)$ is defined as the maximum integer $n$ contained in the number $x$". — Mauro ALLEGRANZA, Jul 23 '20 at 12:17

score 2 · Answer 1 · answered Jul 23 '20 at 12:17

$E(x)$ the floor or the greatest integer function : $E(x)=[x]$, $x=[x]+q$, where $[x]$ is integer $n$ and $q=x-[x]$ is the fractional part. $$S=\int_{0}^{x} [t] dt=\int_{0}^{1} 0 dt+ \int_{1}^{2} 1 dt+\int_{2}^{3} 2 dt+.......+\int_{n-1}^{n} (n-1) dt+\int_{n}^{n+q} n dx$$ $$S=1+2+3+4+...+(n-1)+nq=n(n-1)+nq=n(n-1)/2+nq$$ $$ \implies S=[x]([x]-1)/2+[x](x-[x]).$$

score 2 · Accepted Answer · answered Jul 23 '20 at 12:18

I think it is the floor, $E(x) = \lfloor x\rfloor$. If $x$ is an integer, then you find the expression for the sum of integers, and if $x$ is not an integer, you add the missing part of the rectangle. In French, the notation $E(x)$ is used for the floor since it is called "partie entière".

What is the function $E(x)$?

2 Answers2