If f:R`to`R is function defined by f(x) = `[x]^3 cos ((2x-1)/2)pi` , where [x] denotes the greatest integer function, then f is :

Options

A. discontinuous only at x=0
B. discontinuous only at non-zero integral value of x
C. continuous only at x=0
D. continuous for every real x.

Answer

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

\[ f(x) = \lfloor x \rfloor^3 \cos\left(\frac{2x - 1}{2} \pi\right) \]

where \( \lfloor x \rfloor \) denotes the greatest integer function, we will analyze the function step by step.

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

The function consists of two parts:
1. The greatest integer function \( \lfloor x \rfloor \), which is discontinuous at integer values.
2. The cosine function \( \cos\left(\frac{2x - 1}{2} \pi\right) \), which is continuous everywhere.

### Step 2: Analyze the behavior of \( f(x) \)

We can rewrite the function as:

\[ f(x) = \lfloor x \rfloor^3 \cos\left(\pi x - \frac{\pi}{2}\right) \]

Using the identity \( \cos\left(\frac{\pi}{2} - \theta\right) = \sin(\theta) \), we can express it as:

\[ f(x) = \lfloor x \rfloor^3 \sin(\pi x) \]

### Step 3: Evaluate limits around integers

To check continuity at integer points, we need to evaluate the left-hand limit and the right-hand limit as \( x \) approaches an integer \( n \).

1. **Left-hand limit** as \( x \to n^- \):
- For \( x \) just less than \( n \), \( \lfloor x \rfloor = n - 1 \).
- Thus, \( f(x) = (n - 1)^3 \sin(\pi x) \).
- As \( x \to n^- \), \( \sin(\pi x) \to 0 \).
- Therefore, \( \lim_{x \to n^-} f(x) = (n - 1)^3 \cdot 0 = 0 \).

2. **Right-hand limit** as \( x \to n^+ \):
- For \( x \) just greater than \( n \), \( \lfloor x \rfloor = n \).
- Thus, \( f(x) = n^3 \sin(\pi x) \).
- As \( x \to n^+ \), \( \sin(\pi x) \to 0 \).
- Therefore, \( \lim_{x \to n^+} f(x) = n^3 \cdot 0 = 0 \).

3. **Value of the function at the integer**:
- \( f(n) = n^3 \sin(n \pi) = n^3 \cdot 0 = 0 \).

### Step 4: Conclusion on continuity

Since:

\[ \lim_{x \to n^-} f(x) = 0, \quad \lim_{x \to n^+} f(x) = 0, \quad \text{and} \quad f(n) = 0 \]

we conclude that \( f(x) \) is continuous at every integer \( n \).

### Step 5: General continuity

The function \( f(x) \) is continuous at all points in \( \mathbb{R} \) because:
- The left-hand and right-hand limits are equal to the function value at every integer.
- The cosine and sine functions are continuous everywhere.

Thus, we can conclude that:

\[ \text{The function } f \text{ is continuous for all } x \in \mathbb{R}. \]

### Final Answer

The function \( f \) is continuous for every real \( x \).

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