Is it possible to simulate a floor() function with elementary arithmetic?

Let $$f_1(x)=x+c_1$$ $$f_2(x)=x\cdot c_2$$ where $c_1,c_2$ are some constants, and $f_1$ represents addition and subtraction, and $f_2$ represents multiplication and division. Functions $f_1$ and $f_2$ have inverse function, but $\lfloor{x}\rfloor$ does not have an inverse function. So, if we assume it is possible to find $\lfloor{x}\rfloor$ from $x$ using $f_1$ and $f_2$, then it is possible to find $x$ from $\lfloor{x}\rfloor$. Contradiction.

The floor function has jump discontinuities.

A function obtained with elementary arithmetic is a rational function and those can only have pole (or infinite) discontinuities.

Yes it is possible, but depends on the available arithmetic and the rounding mode. With IEEE-double you can round a double $x$ to the nearest integer with this:

$c=2^{52};$

$x = x + c; \mathrm{round} = x - c$ for $x\ge 0\;$ and

$x = x - c; \mathrm{round} = x + c$ for $x<0.$

With the rounding to the next integer you can implement floor.