Let $f:[0, \infty) \rightarrow[0, \infty)$ be defined as
$\mathrm{f}(\mathrm{x})=\int_{0}^{\mathrm{x}}[\mathrm{y}] \mathrm{dy}$
where $[\mathrm{x}]$ is the greatest integer less than or equal to $\mathrm{x}$. Which of the following is true?
Correct Option: 1
$f:[0, \infty) \rightarrow[0, \infty), f(\mathrm{x})=\int_{0}^{\mathrm{x}}[\mathrm{y}] \mathrm{dy}$
Let $\mathrm{x}=\mathrm{n}+f, f \in(0,1)$
So $f(\mathrm{x})=0+1+2+\ldots+(\mathrm{n}-1)+\int_{\mathrm{n}}^{\mathrm{n}+f} \mathrm{n} \mathrm{dy}$
$f(\mathrm{x})=\frac{\mathrm{n}(\mathrm{n}-1)}{2}+\mathrm{n} f$
$=\frac{[x]([x]-1)}{2}+[x]\{x\}$
Note $\lim _{x \rightarrow n^{+}} f(x)=\frac{n(n-1)}{2}, \lim _{x \rightarrow n^{-}} f(x)=\frac{(n-1)(n-2)}{2}+(n-1)$
$=\frac{\mathrm{n}(\mathrm{n}-\mathrm{l})}{2}$
$f(\mathrm{x})=\frac{\mathrm{n}(\mathrm{n}-1)}{2} \quad\left(\mathrm{n} \in \mathrm{N}_{0}\right)$
so $f(\mathrm{x})$ is cont. $\forall \mathrm{x} \geq 0$ and diff. except at integer points