Algebra Of Complex Numbers

To analyze the function , where denotes the greatest integer less than or equal to , follow these steps:

Discontinuity Analysis

1. Check where may change value within . Notice where is an integer, specifically because .

2. Points where changes occur are . Evaluate continuity at these points:

• For jumps from 0 to 1. At , function switches from 0 to non-zero value. Thus, is discontinuous.
• For jumps from 1 to 2. Similar discontinuity due to value change in .
• For jumps from 2 to 3. Another discontinuity point due to the evaluation of polynomial term.

3. Therefore, is not continuous at these three points.

Continuity and Differentiability

1. The problem asks to find exactly one point where the function is continuous but not differentiable:

• Points: seems continuous upon inspecting polynomial terms where no jumps occur in integers.
• However, at , the function may smoothly transition through polynomial terms, confirming continuity without differentiability due to the modulus of polynomial not being zero.

Thus, function is continuous at this domain but not differentiable.