The greatest integer function $f(x) = [x]$ is differentiable for all values of

The greatest integer function $f(x) = [x]$ gives the largest integer that is less than or equal to $x$.

For example:

• $[2.7] = 2$
• $[5.1] = 5$
• $[3] = 3$
• $[-1.5] = -2$

This function creates a step pattern, staying constant between integers and jumping at each integer.

---

A function is differentiable at a point if it is continuous at that point and has a well-defined derivative.

---

At integer points (like $x = 1, 2, 3, ...$):

Consider $x = 2$:

• For $x = 1.9, 1.99, 1.999...$, we have $[x] = 1$
• At $x = 2$, we have $[2] = 2$
• For $x = 2.01, 2.1, ...$, we have $[x] = 2$

The function jumps from 1 to 2 at $x = 2$, creating a discontinuity.

Since the function is not continuous at integer points, it is not differentiable at these points.

This occurs at every integer.

---

At non-integer points (like $x = 1.5, 2.7, \pi, ...$):

Consider $x = 2.5$:

Between $x = 2$ and $x = 3$, the function is constant:

• $[2.3] = 2$
• $[2.5] = 2$
• $[2.8] = 2$

The function is constant on the interval, so it is continuous with derivative equal to 0.

The function is differentiable at all non-integer points.

---

The greatest integer function is differentiable for all real numbers except integers.

Therefore, $f(x) = [x]$ is differentiable for $x \in \mathbb{R} - \mathbb{Z}$