Give an function which is not metric space 

A function that is **not a metric** fails one or more of the metric space properties: non-negativity, identity of indiscernibles, symmetry, or the triangle inequality. Here’s an example of a function that is not a metric:

Define \( d : \mathbb{R} \times \mathbb{R} \to \mathbb{R} \) by:

\[
d(x, y) = |x - y| + 1
\]

### Reason why this is not a metric:
1. **Non-negativity**: It satisfies non-negativity since \( d(x, y) \geq 0 \).
2. **Identity of indiscernibles**: This is violated because \( d(x, x) = |x - x| + 1 = 1 \neq 0 \). In a metric, \( d(x, x) \) must be 0 for all \( x \).
3. **Symmetry**: It satisfies symmetry since \( d(x, y) = d(y, x) \).
4. **Triangle inequality**: It still satisfies the triangle inequality. However, because it fails the identity of indiscernibles, it is not a metric.

Thus, \( d(x, y) = |x - y| + 1 \) is an example of a function that is **not a metric**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. What are the four properties required for a function to be a metric?
2. Can you give another example of a non-metric function?
3. How do we prove the triangle inequality for a given function?
4. Why is the identity of indiscernibles essential for metric spaces?
5. How would you modify the function \( d(x, y) = |x - y| + 1 \) to make it a valid metric?

**Tip:** When checking if a function is a metric, always start by verifying the identity of indiscernibles property, as it's often the easiest to evaluate!

Explore the concept of non-metric functions in metric spaces with an example function d(x, y) = |x - y| + 1. This function violates the identity of indiscernibles property crucial for metrics. Learn about the properties required for a function to be a metric and why d(x, y) = |x - y| + 1 fails to meet these criteria.

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