* (inclusive) to {@code upper} (inclusive). (These are the same values contained in {@code
   * Range.closed(lower, upper)}.)
   *
   * @throws IllegalArgumentException if {@code lower} is greater than {@code upper}
   * @since 23.0
   */
  public static ContiguousSet<Integer> closed(int lower, int upper) {
    return create(Range.closed(lower, upper), DiscreteDomain.integers());
  }

  /**

                    },
                    {
                        "type": "greater_than_equal",
                        "loc": ["query", "offset"],
                        "msg": "Input should be greater than or equal to 0",
                        "input": "-1",
                        "ctx": {"ge": 0},
                    },
                    {
                        "type": "literal_error",

      size:
        lessThan: "10MiB" # match objects with size less than this value (e.g. 10MiB)
        greaterThan: 1MiB # match objects with size greater than this value (e.g. 1MiB)
      purge:
          # retainVersions: 0 # (default) delete all versions of the object. This option is the fastest.
          # retainVersions: 5 # keep the latest 5 versions of the object.
  

```Python hl_lines="9"
{!../../docs_src/path_params_numeric_validations/tutorial005.py!}
```

## 🔢 🔬: 🎈, 🌘 🌘 & 🌘 🌘

🔢 🔬 👷 `float` 💲.

📥 🌐❔ ⚫️ ▶️️ ⚠ 💪 📣 <abbr title="greater than"><code>gt</code></abbr> &amp; 🚫 <abbr title="greater than or equal"><code>ge</code></abbr>. ⏮️ ⚫️ 👆 💪 🚚, 🖼, 👈 💲 🔜 👑 🌘 `0`, 🚥 ⚫️ 🌘 🌘 `1`.

, `0.5` 🔜 ☑ 💲. ✋️ `0.0` ⚖️ `0` 🔜 🚫.

&amp; 🎏 <abbr title="less than"><code>lt</code></abbr>.

   */
  public double populationCovariance() {
    checkState(count() != 0);
    return sumOfProductsOfDeltas / count();
  }

  /**
   * Returns the sample covariance of the values. The count must be greater than one.
   *
   * <p>This is not guaranteed to return zero when the dataset consists of the same pair of values
   * multiple times, due to numerical errors.
   *
   * <h3>Non-finite values</h3>
   *

constraints.AssertTrue.message  = {item} must be true.
constraints.DecimalMax.message  = {item} must be less than ${inclusive == true ? 'or equal to ' : ''}{value}.
constraints.DecimalMin.message  = {item} must be greater than ${inclusive == true ? 'or equal to ' : ''}{value}.
constraints.Digits.message      = {item} is numeric value out of bounds (<{integer} digits>.<{fraction} digits> expected).

class Item(BaseModel):
    name: str
    description: str | None = Field(
        default=None, title="The description of the item", max_length=300
    )
    price: float = Field(gt=0, description="The price must be greater than zero")
    tax: float | None = None


@app.put("/items/{item_id}")
async def update_item(item_id: int, item: Item = Body(embed=True)):
    results = {"item_id": item_id, "item": item}

class Item(BaseModel):
    name: str
    description: Union[str, None] = Field(
        default=None, title="The description of the item", max_length=300
    )
    price: float = Field(gt=0, description="The price must be greater than zero")
    tax: Union[float, None] = None


@app.put("/items/{item_id}")
async def update_item(item_id: int, item: Item = Body(embed=True)):
    results = {"item_id": item_id, "item": item}

                    },
                    {
                        "type": "greater_than_equal",
                        "loc": ["query", "offset"],
                        "msg": "Input should be greater than or equal to 0",
                        "input": "-1",
                        "ctx": {"ge": 0},
                    },
                    {
                        "type": "literal_error",

class Item(BaseModel):
    name: str
    description: Union[str, None] = Field(
        default=None, title="The description of the item", max_length=300
    )
    price: float = Field(gt=0, description="The price must be greater than zero")
    tax: Union[float, None] = None


@app.put("/items/{item_id}")
async def update_item(item_id: int, item: Annotated[Item, Body(embed=True)]):
    results = {"item_id": item_id, "item": item}