From the graph of the function $f(x)=|x-a|$, we can see that when $a \leqslant 1$, this function is monotonically increasing on $[1,+\infty)$. Therefore, “$a=1$” is a sufficient but not necessary condition for “the function $f(x)=|x-a|$ is increasing on the interval $[1,+\infty)$”. The answer is $\mathrm{A}$.
(2) (2006 - Hunan) “ $a=1$ ” is “the function $f(x)=|x-a|$ is increasing on the interval $[1,+\infty)$ ” ( ).
(A) a sufficient but not necessary condition
(B) a necessary but not sufficient condition
(C) a necessary and sufficient condition
(D) neither a sufficient nor a necessary condition