First we have to prove that the usual absolute value function f(x) = | x | is continuous. But this is clear from the graph of that function (nonetheless, can you prove it formally ?).
Now the statement follows immediately from the fact that the composition of two continuous functions yields another continuous function.
As in the case of proving that a polynomial is continuous, the abstract case is much easier to prove that trying to prove, say, that the absolute value of a particular polynomial is continuous.