3.2.47 Show that the absolute-value function
: ℝℝ is continuous.
Let 𝑓: ℝℝ be the absolute value function, 𝑓 𝑥 = 𝑥 .
Let (𝑎, 𝑏) be an open interval on the range of 𝑥 .
There are three possibilities for 𝑎, 𝑏 .
case 1: 𝑎 < 𝑏 ≤ 0
Then 𝑓 −1 𝑎, 𝑏 = . Note that  is an open set.
case 2: 𝑎 < 0 < 𝑏
Then 𝑓 −1 𝑎, 𝑏 = (−𝑏, 𝑏). Note (−𝑏, 𝑏) is an open set.
case 3: 0 ≤ 𝑎 < 𝑏
Then 𝑓 −1 𝑎, 𝑏 = (−b, −a)⋃(a, b) which is also an open set.
So for every open interval 𝑎, 𝑏 in the range of the absolute value function, the inverse
image 𝑈 = 𝑓 −1 𝑎, 𝑏 is an open set, where 𝑓 𝑈 ⊆ (𝑎, 𝑏). Therefore, the absolute value
function is continuous.