Example 1 Determine the periodicity of the function $f(x)=x-[x], x \in \mathbf{R}$ (where $[x]$ denotes the greatest integer not exceeding $x$) and sketch its graph.
Solve As shown in Figure $1-1-3$, we draw the graph of $f(x)$.
From the graph of $f(x)$, we can see that when $x \in \mathbf{R}$, $f(x)=x-[x]$ is a periodic function, and $T=1$ is one of its positive periods. In fact, for $x \in$ $\mathbf{R}$, we have
$$
\begin{aligned}
f(x+1) & =x+1-[x+1]=x+1-[x]-1 \\
& =x-[x]=f(x) .
\end{aligned}
$$

Explanation In fact, $T=1$ is the smallest positive period of $f(x)$. We can prove this by contradiction: If $l(0<l<1)$ is a period of $f(x)$, then for any $x \in \mathbf{R}$, we have:
$$
f(x+l)=f(x),
$$

Let $x=0$, we get $f(l)=f(0)=0$. This contradicts $f(l)=l-[l]=l \neq 0$. Therefore, any $l(0<l<1)$ is not a positive period of $f(x)$.