If ` f: R to R ` is function defined by ` f(x) = [x-1] cos ( (2x -1)/(2)) pi`, where [.] denotes the greatest integer function , then f is :

Options

A. discontinuous at all integral values of x except at x=1
B. Continuous only at x=1
C. continous for every real x
D. discontinuous only at x=1

Answer

To determine the continuity of the function \( f: \mathbb{R} \to \mathbb{R} \) defined by

\[ f(x) = \lfloor x - 1 \rfloor \cos\left(\frac{(2x - 1)}{2} \pi\right), \]

where \(\lfloor . \rfloor\) denotes the greatest integer function, we will analyze the components of the function.

### Step 1: Analyze the components of the function

1. **Greatest Integer Function**: The function \(\lfloor x - 1 \rfloor\) is discontinuous at integer values of \(x\), specifically at \(x = n + 1\) for \(n \in \mathbb{Z}\). This is because the greatest integer function jumps at these points.

2. **Cosine Function**: The function \(\cos\left(\frac{(2x - 1)}{2} \pi\right)\) is continuous everywhere since cosine is a continuous function for all real numbers.

### Step 2: Determine the overall continuity of \(f(x)\)

The product of two functions is continuous if both functions are continuous. However, since \(\lfloor x - 1 \rfloor\) is discontinuous at integer points, we need to check the behavior of \(f(x)\) at these points.

### Step 3: Check the function at integer points

Let \(x = n + 1\) where \(n\) is an integer. At this point:

- \(\lfloor x - 1 \rfloor = \lfloor n + 1 - 1 \rfloor = \lfloor n \rfloor = n\).
- The cosine term becomes \(\cos\left(\frac{(2(n + 1) - 1)}{2} \pi\right) = \cos\left(\frac{(2n + 2 - 1)}{2} \pi\right) = \cos\left(\frac{(2n + 1)}{2} \pi\right)\).

### Step 4: Behavior around integer points

As \(x\) approaches \(n + 1\) from the left (\(x \to n + 1^-\)):
- \(\lfloor x - 1 \rfloor = n\)
- \(\cos\left(\frac{(2(n + 1) - 1)}{2} \pi\right)\) will be evaluated at a point slightly less than \(n + 1\).

As \(x\) approaches \(n + 1\) from the right (\(x \to n + 1^+\)):
- \(\lfloor x - 1 \rfloor = n\)
- The cosine term will also be evaluated at a point slightly more than \(n + 1\).

### Step 5: Conclusion about continuity

Since the cosine function is continuous and the greatest integer function causes a jump at integer values, the overall function \(f(x)\) is discontinuous at \(x = n + 1\) for any integer \(n\). However, for all other real numbers, \(f(x)\) is continuous.

Thus, we conclude that:

- \(f(x)\) is continuous for all \(x \in \mathbb{R}\) except at points where \(x\) is an integer.

### Final Answer

The function \(f\) is discontinuous at integer points and continuous elsewhere.

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