Examine the derivability of the following functions at the specified points :
`[x]" at "x = 1`

Answer

To examine the derivability of the greatest integer function \( f(x) = [x] \) at the point \( x = 1 \), we will follow these steps:

### Step 1: Define the Greatest Integer Function
The greatest integer function, denoted as \( [x] \), gives the largest integer less than or equal to \( x \).

### Step 2: Determine the Value of the Function at \( x = 1 \)
At \( x = 1 \):
\[ f(1) = [1] = 1 \]

### Step 3: Calculate the Left-Hand Limit \( f(1^-) \)
The left-hand limit as \( x \) approaches 1 from the left is:
\[ f(1^-) = [1^-] = [0.999...] = 0 \]

### Step 4: Calculate the Right-Hand Limit \( f(1^+) \)
The right-hand limit as \( x \) approaches 1 from the right is:
\[ f(1^+) = [1^+] = [1.000...] = 1 \]

### Step 5: Check Continuity at \( x = 1 \)
For a function to be derivable at a point, it must first be continuous at that point. We check if:
\[ f(1^-) \neq f(1) \quad \text{and} \quad f(1^-) \neq f(1^+) \]
From our calculations:
- \( f(1^-) = 0 \)
- \( f(1) = 1 \)
- \( f(1^+) = 1 \)

Since \( f(1^-) \neq f(1) \), the function is not continuous at \( x = 1 \).

### Step 6: Conclusion on Derivability
Since the function is not continuous at \( x = 1 \), it cannot be derivable at that point. Therefore, we conclude:
\[ \text{The function } f(x) = [x] \text{ is non-derivable at } x = 1. \]

---