Let $f(x)=\left\{\begin{array}{cl}x^2 \sum\limits_{r=0}^{\left[\frac{1}{|x|}\right]} r & , x \neq 0\\ k & , x=0\end{array}\right.$ where [.] denotes the greatest integer function. The value of k for which f is continuous at x = 0, is __________

Clearly, $f(x)$ is an even function and is given by

$f(x)=\left\{\begin{array}{c} \frac{x^2}{2}\left[\frac{1}{|x|}\right]\left(\left[\frac{1}{|x|}\right]+1\right), &x \neq 0 \\ k~~~~~~ , &x=0\end{array}\right.$

If f is continuous at x = 0, then

$\lim\limits_{x \rightarrow 0} f(x)=f(0)$

$\Rightarrow \lim\limits_{x \rightarrow 0} \frac{x^2}{2}\left[\frac{1}{|x|}\right]\left(\left[\frac{1}{|x|}\right]+1\right)=k$

$\Rightarrow \lim\limits_{x \rightarrow 0} \frac{\left[\frac{1}{|x|}\right]\left(\left[\frac{1}{|x|}\right]+1\right)}{\frac{1}{|x|^2}}=2 k$

$\Rightarrow \lim\limits_{y \rightarrow \infty} \frac{[y]([y]+1)}{y^2}=2 k$, where $y=\frac{1}{|x|}$

$\Rightarrow \lim\limits_{y \rightarrow \infty} \frac{[y]}{y}\left(\frac{[y]}{y}+\frac{1}{y}\right)=2 k$

$\Rightarrow 1(1+0)=2 k$             $\left[∵ \lim\limits_{x \rightarrow \infty} \frac{x}{[x]}=1\right]$

$\Rightarrow k=1$