Prove that the greatest integer function defined by f(x) = [x], 0<x<3 is not differntiable at x = 1 and x = 2

To prove that the greatest integer function \( f(x) = \lfloor x \rfloor \), defined on the interval \( 0 < x < 3 \), is **not differentiable** at \( x = 1 \) and \( x = 2 \), we need to check the differentiability conditions at those points.

### 1. Understanding the Greatest Integer Function:

The greatest integer function (or floor function) \( f(x) = \lfloor x \rfloor \) gives the largest integer less than or equal to \( x \). For example:
- \( \lfloor 1.2 \rfloor = 1 \)
- \( \lfloor 2.9 \rfloor = 2 \)

### 2. Differentiability at \( x = 1 \) and \( x = 2 \):

To check whether \( f(x) \) is differentiable at \( x = 1 \) and \( x = 2 \), we must first check the continuity and the behavior of the function around these points.

#### (a) At \( x = 1 \):

- **For \( x \in (0, 1) \)**:  
  If \( 0 < x < 1 \), then \( \lfloor x \rfloor = 0 \).

- **At \( x = 1 \)**:  
  \( \lfloor 1 \rfloor = 1 \).

- **For \( x \in (1, 2) \)**:  
  If \( 1 < x < 2 \), then \( \lfloor x \rfloor = 1 \).

Thus, there is a **jump** at \( x = 1 \), from \( 0 \) (as \( x \to 1^- \)) to \( 1 \) (at \( x = 1 \)), indicating a discontinuity.

Since differentiability implies continuity, and \( f(x) \) is **not continuous at \( x = 1 \)**, it is not differentiable at \( x = 1 \).

#### (b) At \( x = 2 \):

- **For \( x \in (1, 2) \)**:  
  If \( 1 < x < 2 \), then \( \lfloor x \rfloor = 1 \).

- **At \( x = 2 \)**:  
  \( \lfloor 2 \rfloor = 2 \).

- **For \( x \in (2, 3) \)**:  
  If \( 2 < x < 3 \), then \( \lfloor x \rfloor = 2 \).

There is again a **jump** at \( x = 2 \), from \( 1 \) (as \( x \to 2^- \)) to \( 2 \) (at \( x = 2 \)). Hence, the function is **not continuous at \( x = 2 \)**.

Therefore, \( f(x) \) is **not differentiable at \( x = 2 \)** either.

### 3. Conclusion:

The greatest integer function \( f(x) = \lfloor x \rfloor \) is not differentiable at both \( x = 1 \) and \( x = 2 \) because the function exhibits jumps at these points, making it discontinuous and hence non-differentiable.

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Would you like more details or have any questions?

Here are 5 related questions to expand this discussion:

1. How does the concept of continuity relate to differentiability?
2. What are other examples of functions that are not differentiable at certain points?
3. How would we compute the left-hand and right-hand limits for differentiability tests?
4. What is the difference between the floor function and the ceiling function?
5. How can differentiability be defined rigorously using limits?

**Tip**: A function must be continuous at a point for it to be differentiable at that point, but continuity alone does not guarantee differentiability.

Prove that the greatest integer function defined by f(x) = [x], 0<x<3 is not differentiable at x = 1 and x = 2

Explore the non-differentiability of the greatest integer function f(x) = ⌊x⌋ within the interval 0 < x < 3. This problem analyzes differentiability at x = 1 and x = 2, highlighting discontinuities and jumps in the floor function. Suitable for advanced calculus students in Grades 11-12.

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