The function `f(x)=x-[x]` , where `[]` denotes the greatest integer function is (a) continuous everywhere (b) continuous at integer points only (c) continuous at non-integer points only (d) differentiable everywhere

Explanation

To determine the continuity and differentiability of the function \( f(x) = x - [x] \), where \( [x] \) denotes the greatest integer function, we will analyze the function step by step.

### Step 1: Understanding the Function
The function \( f(x) = x - [x] \) represents the fractional part of \( x \). This means that for any real number \( x \), \( f(x) \) gives the decimal part of \( x \). For example:
- If \( x = 2.3 \), then \( [x] = 2 \) and \( f(2.3) = 2.3 - 2 = 0.3 \).
- If \( x = 3.7 \), then \( [x] = 3 \) and \( f(3.7) = 3.7 - 3 = 0.7 \).
- If \( x = 5 \), then \( [x] = 5 \) and \( f(5) = 5 - 5 = 0 \).

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