`f(x)=[x] and g(x)={{:(1",",x gt1),(2",", x le 1):}`, (where `[.]` represents the greatest integer function).
Then `g(f(x))` is discontinuous at x = _______________

Answer

To solve the problem, we need to analyze the functions \( f(x) \) and \( g(x) \) step by step.

### Step 1: Understand the function \( f(x) \)
The function \( f(x) = [x] \) represents the greatest integer function, which means it gives the largest integer less than or equal to \( x \).

- For example:
- \( f(1.5) = 1 \)
- \( f(2.3) = 2 \)
- \( f(2) = 2 \)
- \( f(3) = 3 \)

### Step 2: Understand the function \( g(x) \)
The function \( g(x) \) is defined piecewise:
- \( g(x) = 1 \) if \( x > 1 \)
- \( g(x) = 2 \) if \( x \leq 1 \)

### Step 3: Find \( g(f(x)) \)
Now we need to find \( g(f(x)) \):
- If \( x > 1 \), then \( f(x) = [x] \) will be at least \( 1 \).
- If \( 1 < x < 2 \), then \( f(x) = 1 \) and \( g(f(x)) = g(1) = 2 \).
- If \( x \geq 2 \), then \( f(x) \) will be \( 2 \) or more, and \( g(f(x)) = g(2) = 1 \).

- If \( x \leq 1 \), then \( f(x) = [x] \) will be \( 0 \) or \( 1 \).
- If \( x < 1 \), then \( f(x) = 0 \) and \( g(f(x)) = g(0) = 2 \).
- If \( x = 1 \), then \( f(x) = 1 \) and \( g(f(x)) = g(1) = 2 \).

### Step 4: Summarize \( g(f(x)) \)
From the above analysis, we can summarize \( g(f(x)) \) as follows:
- For \( x < 1 \): \( g(f(x)) = 2 \)
- For \( 1 \leq x < 2 \): \( g(f(x)) = 2 \)
- For \( x \geq 2 \): \( g(f(x)) = 1 \)

### Step 5: Identify points of discontinuity
The function \( g(f(x)) \) changes from \( 2 \) to \( 1 \) at \( x = 2 \). Therefore, we need to check the continuity at this point:
- As \( x \) approaches \( 2 \) from the left (\( x \to 2^- \)), \( g(f(x)) = 2 \).
- As \( x \) approaches \( 2 \) from the right (\( x \to 2^+ \)), \( g(f(x)) = 1 \).

Since the left-hand limit and the right-hand limit at \( x = 2 \) are not equal, \( g(f(x)) \) is discontinuous at \( x = 2 \).

### Final Answer
Thus, \( g(f(x)) \) is discontinuous at \( x = 2 \).

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