Show that the function `f` defined by `f(x)=|1-x+|x||` is everywhere continuous.

Explanation

To show that the function \( f \) defined by \( f(x) = |1 - x + |x|| \) is everywhere continuous, we will analyze the function by breaking it down into different cases based on the values of \( x \).

### Step 1: Understanding the function
The function involves absolute values, so we need to consider different cases for \( x \):
1. When \( x \geq 0 \)
2. When \( x < 0 \)

### Step 2: Case 1: \( x \geq 0 \)
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