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Let f(x)=[x]{x^(2)}+[x][x^(2)]+{x}[x^(2)...

Let `f(x)=[x]{x^(2)}+[x][x^(2)]+{x}[x^(2)]+{x}{x^(2)}, AA x in [0, 10]`
`[.] and {.}` the greatest integer and fractional part functions respectively). The number of points of discontinuity of `f(x)` is

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To solve the problem, we need to analyze the function \( f(x) \) defined as: \[ f(x) = [x]\{x^2\} + [x][x^2] + \{x\}[x^2] + \{x\}\{x^2\} \] where \( [.] \) denotes the greatest integer function (floor function) and \( \{.\} \) denotes the fractional part function. ### Step 1: Understanding the components of \( f(x) \) 1. **Greatest Integer Function**: For any real number \( x \), \( [x] \) is the largest integer less than or equal to \( x \). 2. **Fractional Part Function**: The fractional part \( \{x\} \) is defined as \( x - [x] \). ### Step 2: Rewrite \( f(x) \) We can express \( x \) as: \[ x = [x] + \{x\} \] Similarly, for \( x^2 \): \[ x^2 = [x^2] + \{x^2\} \] ### Step 3: Substitute into \( f(x) \) Using the above definitions, we can rewrite \( f(x) \): \[ f(x) = [x]\{x^2\} + [x][x^2] + \{x\}[x^2] + \{x\}\{x^2\} \] ### Step 4: Analyze the points of discontinuity The points of discontinuity for \( f(x) \) will occur at points where either \( [x] \) or \( \{x\} \) changes, which happens at integer values of \( x \). 1. **Discontinuities of \( [x] \)**: The function \( [x] \) is discontinuous at every integer point. 2. **Discontinuities of \( \{x\} \)**: The function \( \{x\} \) is continuous everywhere except at integer points where it jumps from 1 back to 0. ### Step 5: Identify integer points in the interval [0, 10] The integer points in the interval [0, 10] are: \[ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \] This gives us a total of 11 points. ### Step 6: Conclusion Since \( f(x) \) is a combination of continuous functions except at the integer points, we conclude that \( f(x) \) is discontinuous at these 11 points. Thus, the number of points of discontinuity of \( f(x) \) is: \[ \text{Number of points of discontinuity} = 11 \] ### Final Answer The number of points of discontinuity of \( f(x) \) is \( 11 \). ---
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