Question

A function \( f(x) = |1 - x + |x|| \) is:

• A. discontinuous at \( x = 1 \) only
• B. discontinuous at \( x = 0 \) only
• C. discontinuous at \( x = 0, 1 \)
• D. continuous everywhere

Hint:

For continuity analysis, evaluate left-hand and right-hand limits at critical points.

Answer 1

The Correct Option is D

Step 1: Analyze the given function. 1. **Case 1:** For \( x \geq 0 \), \( |x| = x \). Then: \[ f(x) = |1 - x + x| = |1| = 1. \] 2. **Case 2:** For \( x<0 \), \( |x| = -x \). Then: \[ f(x) = |1 - x - x| = |1 - 2x|. \] Step 2: Check continuity. For \( x \geq 0 \), \( f(x) = 1 \). For \( x<0 \), \( f(x) = |1 - 2x| \). At the transition point \( x = 0 \): \[ f(0^+) = 1, \quad f(0^-) = |1 - 2(0)| = 1. \] Similarly, at \( x = 1 \), \( f(1^+) = 1 \) and \( f(1^-) = 1 \). Thus, \( f(x) \) is continuous everywhere.
Final Answer: \( \boxed{ {(D)}} \)