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.