Let f:[0,)[0,) be defined as f(x)=_0^x[y],dy, where [x] is the greatest integer x.

Question

Let $f:[0,\infty)	o[0,\infty)$ be defined as $f(x)=\int_0^x[y]\,dy$, where $[x]$ is the greatest integer $\le x$. Which of the following is true?

Accepted Answer

Writing $x=I+f$ with $I=[x]$, $0\le f<1$:
$$f(x)=\frac{I(I-1)}{2}+I\cdot f=\frac{[x]([x]-1)}{2}+[x](x-[x])$$
This is continuous everywhere (as an accumulation via a continuous integral of a piecewise-constant function). Checking derivatives at an integer $x=n$: the left derivative equals $n-1$ (the value of $[x]$ just below $n$) and the right derivative equals $n$; these differ, so $f$ is not differentiable at integers but is continuous everywhere.

Solutions · 1