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 \( f(x) = |x| + |x - 1| \) is defined using absolute values.

Case-wise analysis:

• For \( x \geq 1 \): \( f(x) = x + (x - 1) = 2x - 1 \). This expression is continuous and differentiable.
• For \( 0 \leq x < 1 \): \( f(x) = x + (1 - x) = 1 \). This expression is continuous but not differentiable at the boundary points.
• For \( x < 0 \): \( f(x) = -x + (1 - x) = 1 - 2x \). This expression is continuous and differentiable.

Conclusion:
The function \( f(x) \) is continuous everywhere but not differentiable at \( x = 0 \) and \( x = 1 \).