Step 1: Analyze the function \( f(x) = |1 - x + |x|| \) by considering two cases.
Case 1: If \( x \geq 0 \), then \( |x| = x \). The function simplifies to \( f(x) = |1 - x + x| = |1| = 1 \).
Case 2: If \( x<0 \), then \( |x| = -x \). The function becomes \( f(x) = |1 - x - x| = |1 - 2x| \).
Step 2: Check the continuity of the function.
For \( x \geq 0 \), \( f(x) = 1 \), which is continuous.
For \( x<0 \), \( f(x) = |1 - 2x| \), which is continuous as it's a piecewise linear function.
At \( x = 0 \), the left-hand limit (LHL) and right-hand limit (RHL) are calculated as follows:
LHL = \( f(0^-) = |1 - 2(0)| = 1 \)
RHL = \( f(0^+) = 1 \)
Since LHL = RHL, the function is continuous at \( x = 0 \).
Step 3: Conclude the result.
The function \( f(x) \) is continuous for all \( x \), therefore it is continuous everywhere.