# Piecewise Functions - 3

**Calculus**Level 5

Let a function \(f : \mathbb{R} \rightarrow \mathbb{R}\) be defined as:

\[f(x) = \begin{cases} \sin x, & \text{if } a < x \leq b \\ |x - c | - d, & \text{if } x \leq a \text{ or } x > b. \end{cases}\]

It is given:

- \(a, b, c, d\) are real numbers
- \(f\) is a differentiable function for all real values of \(x\)

Find the minimum possible value of \(\lvert \lfloor a + b + c + d \rfloor \rvert\).

**Details and assumptions**

\(\lfloor \cdot \rfloor\) is the floor function

This problem is part of the set - Piecewise-defined Functions