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

Answer: (C)

continuous everywhere and differentiable at all points except at x = 0

Answer 2

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 $x = 1$.