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: If \( x \geq 0 \), then \( |x| = x \). Consequently:\[f(x) = |1 - x + x| = |1| = 1.\]2. Case 2: If \( x<0 \), then \( |x| = -x \). Consequently:\[f(x) = |1 - x - x| = |1 - 2x|.\]Step 2: Verify continuity.For \( x \geq 0 \), \( f(x) = 1 \). For \( x<0 \), \( f(x) = |1 - 2x| \). At the critical point \( x = 0 \):\[f(0^+) = 1, \quad f(0^-) = |1 - 2(0)| = 1.\]Similarly, at \( x = 1 \), \( f(1^+) = 1 \) and \( f(1^-) = 1 \). Therefore, \( f(x) \) is continuous for all values of \( x \).
Final Answer: \( \boxed{ {(D)}} \)