(11) Given that $f(x)$ is an even function, when $x \leqslant 0$, $f(x)=x-[x]$ (the symbol $[x]$ represents the greatest integer not exceeding $x$), if the equation $f(x)=k x+k$ ($k>0$) has exactly three distinct real roots, then the range of the real number $k$ is $\qquad$
Solution: $\left[\frac{1}{3}, \frac{1}{2}\right)$
Draw the graphs of the functions $y=f(x)$ and $y=k x+k$ (as shown in the figure).
It is easy to see that the line $y=k x+k$ always passes through the point $(-1,0)$, and $x=-1$ is a root of the equation $f(x)=k x+k$.
From the graph, we can see that,
when $\frac{1-0}{2-(-1)} \leqslant k<\frac{1-0}{1-(-1)}$, i.e., $\frac{1}{3} \leqslant k<\frac{1}{2}$, the graphs of the two functions intersect at exactly three different points.
$\therefore$ The range of values for $k$ is $\left[\frac{1}{3}, \frac{1}{2}\right)$.