To solve the problem, we need to analyze the function \( f(x) = \cos(\pi (|x| + [x])) \), where \([x]\) denotes the greatest integer function (also known as the floor function). We will check the behavior of this function in the intervals \([-1, 0)\) and \([0, 1)\) and determine its continuity and differentiability.
### Step 1: Analyze the function in the interval \([-1, 0)\)
For \( x \) in the interval \([-1, 0)\):
- The absolute value \( |x| = -x \) (since \( x \) is negative).
- The greatest integer function \([x] = -1\) (since \( x \) is in the range \([-1, 0)\)).
Thus, we can write:
\[
f(x) = \cos(\pi (-x - 1)) = \cos(-\pi x - \pi).
\]
Using the property of cosine, \( \cos(-\theta) = \cos(\theta) \), we have:
\[
f(x) = \cos(\pi x + \pi) = -\cos(\pi x).
\]
### Step 2: Analyze the function in the interval \([0, 1)\)
For \( x \) in the interval \([0, 1)\):
- The absolute value \( |x| = x \).
- The greatest integer function \([x] = 0\) (since \( x \) is in the range \([0, 1)\)).
Thus, we can write:
\[
f(x) = \cos(\pi (x + 0)) = \cos(\pi x).
\]
### Step 3: Check continuity at \( x = 0 \)
To check the continuity of \( f(x) \) at \( x = 0 \), we need to evaluate the left-hand limit and the right-hand limit as \( x \) approaches 0.
1. **Left-hand limit**:
\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} -\cos(\pi x) = -\cos(0) = -1.
\]
2. **Right-hand limit**:
\[
\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} \cos(\pi x) = \cos(0) = 1.
\]
Since \( \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x) \), we conclude that \( f(x) \) is discontinuous at \( x = 0 \).
### Step 4: Check differentiability
Since \( f(x) \) is discontinuous at \( x = 0 \), it cannot be differentiable at that point. However, it is differentiable in the intervals \([-1, 0)\) and \([0, 1)\) separately.
### Conclusion
From the analysis, we find that the function \( f(x) \) is not continuous at \( x = 0 \), and hence it is not differentiable at that point. Therefore, the statement that is not true is that \( f(x) \) is continuous at \( x = 0 \).
### Final Answer
The option that is not true is:
- \( f(x) \) is continuous at \( x = 0 \).
