Discuss the differentiability of `f(x)=|x-1|+|x-2|`

Explanation

To discuss the differentiability of the function \( f(x) = |x - 1| + |x - 2| \), we will analyze the function piecewise based on the critical points where the absolute value expressions change, which are at \( x = 1 \) and \( x = 2 \).

### Step 1: Identify the intervals
The function \( f(x) \) can be expressed differently depending on the value of \( x \):
- For \( x < 1 \): Both \( |x - 1| \) and \( |x - 2| \) are negative, so:
\[ f(x) = -(x - 1) - (x - 2) = -x + 1 - x + 2 = -2x + 3 \]
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