# Let f : [ 0 , ∞ ) → [ 0 , ∞ ) be defined as

Question:

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?

1. $f$ is continuous at every point in $[0, \infty)$ and differentiable except at the integer points.

2. $\mathrm{f}$ is both continuous and differentiable except at the integer points in $[0, \infty)$.

3. f is continuous everywhere except at the integer points in $[0, \infty)$.

4. f is differentiable at every point in $[0, \infty)$.

Correct Option: 1

Solution:

$f:[0, \infty) \rightarrow[0, \infty), f(\mathrm{x})=\int_{0}^{\mathrm{x}}[\mathrm{y}] \mathrm{dy}$

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