The function `f(x)=|x-1|+|x-2| is

Options

A. continuous and differentiable everywhere
B. continuous at x=1,2, but differentiable anywhere
C. continuous everywhere, but not differentiable at x=1,2
D. None of the above

Answer

To analyze the function \( f(x) = |x - 1| + |x - 2| \), we will determine its behavior in different intervals based on the critical points where the expressions inside the absolute values change sign.

### Step 1: Identify critical points
The critical points occur when the expressions inside the absolute values equal zero:
- \( x - 1 = 0 \) gives \( x = 1 \)
- \( x - 2 = 0 \) gives \( x = 2 \)