Question

The function \( f(x) = |x| + |x - 2| \) is

• A. continuous, but not differentiable at \( x = 0 \) and \( x = 2 \)
• B. differentiable but not continuous at \( x = 0 \) and \( x = 2 \)
• C. continuous but not differentiable at \( x = 0 \) only
• D. neither continuous nor differentiable at \( x = 0 \) and \( x = 2 \)

Hint:

For absolute value functions, check the points where the function changes form, as they can cause non-differentiability.

Answer 1

The Correct Option is A

We are given the function \( f(x) = |x| + |x - 2| \). 
First, we check the continuity at the points where \( x = 0 \) and \( x = 2 \). At \( x = 0 \), the function is continuous because the left and right limits match. 
However, the function is not differentiable at \( x = 0 \) because the slope changes abruptly from negative to positive. At \( x = 2 \), the function is continuous as both left and right limits match. 
However, the function is not differentiable at \( x = 2 \) due to an abrupt change in slope. 
Thus, the function is continuous but not differentiable at both points \( x = 0 \) and \( x = 2 \).