Question

The function f(x) = |x| + |1 − x| is:

• A. continuous and differentiable at x = 0 only
• B. continuous at x = 0 but nowhere differentiable
• C. continuous everywhere and differentiable at all points except at x = 0
• D. continuous but not differentiable at x = 1

Answer 1

The Correct Option is C

The function \( f(x) = |x| + |1 - x| \) is the sum of two absolute value functions, which are continuous everywhere. However, absolute value functions are not differentiable at the points where their arguments are zero. Specifically:

• \(|x|\) is not differentiable at \(x = 0\).
• \(|1 - x|\) is not differentiable at \(x = 1\).

Thus, \(f(x)\) is continuous everywhere but differentiable at all points except at \(x = 0\) and \(x = 1\).