Discuss the continuity of 'f' in [0, 2], where `f(x) = {{:(|4x - 5|[x],"for",x gt 1),([cos pi x],"for",x le 1):}`, where [x] is greastest integer not greater than x.

Answer

To discuss the continuity of the function \( f \) defined as:

\[ f(x) = \begin{cases} |4x - 5| \cdot [x] & \text{for } x > 1 \\ \cos(\pi x) & \text{for } x \leq 1 \end{cases} \]

on the interval \([0, 2]\), we need to check the continuity at all points in this interval, particularly at the point where the definition of \( f(x) \) changes, which is at \( x = 1 \).

### Step 1: Check continuity for \( x < 1 \)

For \( x \leq 1 \), the function is defined as \( f(x) = \cos(\pi x) \).

1. **Evaluate \( f(0) \)**:
\[ f(0) = \cos(0) = 1 \]

2. **Evaluate \( f(1) \)**:
\[ f(1) = \cos(\pi \cdot 1) = \cos(\pi) = -1 \]

### Step 2: Check continuity for \( x > 1 \)

For \( x > 1 \), the function is defined as \( f(x) = |4x - 5| \cdot [x] \).

1. **Evaluate \( f(2) \)**:
\[ f(2) = |4 \cdot 2 - 5| \cdot [2] = |8 - 5| \cdot 2 = 3 \cdot 2 = 6 \]

### Step 3: Check continuity at \( x = 1 \)

To check continuity at \( x = 1 \), we need to find the left-hand limit, the right-hand limit, and the value of the function at that point.

1. **Left-hand limit as \( x \) approaches 1**:
\[ \lim_{x \to 1^-} f(x) = f(1) = -1 \]

2. **Right-hand limit as \( x \) approaches 1**:
\[ \lim_{x \to 1^+} f(x) = |4 \cdot 1 - 5| \cdot [1] = |4 - 5| \cdot 1 = 1 \cdot 1 = 1 \]

3. **Value of the function at \( x = 1 \)**:
\[ f(1) = -1 \]

### Step 4: Conclusion about continuity

For \( f \) to be continuous at \( x = 1 \), the following must hold:
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) \]

However, we found:
- \( \lim_{x \to 1^-} f(x) = -1 \)
- \( \lim_{x \to 1^+} f(x) = 1 \)
- \( f(1) = -1 \)

Since the left-hand limit and right-hand limit are not equal, \( f(x) \) is not continuous at \( x = 1 \).

Thus, the function \( f(x) \) is continuous on the intervals \([0, 1)\) and \((1, 2]\) but not continuous at \( x = 1 \).

### Final Answer

The function \( f \) is continuous on the interval \([0, 2]\) except at \( x = 1 \).

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