Question

If \( f(x) = |x| + |x - 1| \), then which of the following is correct?

• A. \( f(x) \) is both continuous and differentiable, at \( x = 0 \) and \( x = 1 \)
• B. \( f(x) \) is differentiable but not continuous, at \( x = 0 \) and \( x = 1 \)
• C. \( f(x) \) is continuous but not differentiable, at \( x = 0 \) and \( x = 1 \)
• D. \( f(x) \) is neither continuous nor differentiable, at \( x = 0 \) and \( x = 1 \)

Hint:

Check for points where the function has a change in direction, such as at \( x = 0 \) and \( x = 1 \), which typically leads to discontinuity or non-differentiability.

Answer 1

The Correct Option is C

The function \( f(x) = |x| + |x - 1| \) consists of absolute value functions. - For \( x \geq 1 \), \( f(x) = x + (x - 1) = 2x - 1 \), which is continuous and differentiable. - For \( 0 \leq x<1 \), \( f(x) = x + (1 - x) = 1 \), which is continuous but not differentiable at \( x = 0 \). Thus, \( f(x) \) is continuous but not differentiable at both \( x = 0 \) and \( x = 1 \).