docs: apply a registered Function version in the Function guide

The guide applied the local decorated callable to `table["value"]`, which
raises TypeError: a table is not subscriptable, and add_columns takes an
application of a registered version, not the callable's result.

The guide now walks one example from definition to filled column: define
`double`, call it locally, connect to Enterprise, create a table whose
`value` column matches the parameter's type and nullability, register the
function, apply `version(value=col("value"))`, refresh, wait, and read the
values back. Its snippets come from docs/web-tests/py/test_function.py.

The local tests in that file check what `@udf` sends for registration, how
a registered version binds a column, and that a local connection refuses
both registration and Function columns. The Enterprise walkthrough runs
only when LANCEDB_URI, LANCEDB_API_KEY and LANCEDB_HOST_OVERRIDE are set.

The page also stops promising that a retry after a failed refresh costs
only the unfinished rows.

docs/web-tests/py/test_function.py

@@ -0,0 +1,193 @@
+# SPDX-License-Identifier: Apache-2.0
+# SPDX-FileCopyrightText: Copyright The LanceDB Authors
+
+"""Examples for docs/web/management/function.mdx.
+
+Registering a Function and filling its column need a LanceDB Enterprise
+deployment, so `test_function_enterprise` runs only when the connection
+variables the Enterprise examples already use are set: LANCEDB_URI,
+LANCEDB_API_KEY and LANCEDB_HOST_OVERRIDE, plus LANCEDB_REGION if the
+deployment needs one. It creates the table `function_demo` and a version of
+the Function `double` in that database, and drops both when it finishes.
+
+The other tests run locally and check the client side of the same code: what
+`@udf` sends for registration, how a registered version binds a column, and
+where a local connection stops. They do not run a Function remotely.
+"""
+
+import json
+import os
+import sys
+from pathlib import Path
+
+import lancedb
+import pytest
+from lancedb import col
+from lancedb.functions import FunctionApplication, FunctionVersion
+
+# --8<-- [start:function_define]
+from lancedb import udf
+
+
+@udf
+def double(value: float) -> float:
+    return value * 2
+
+
+# --8<-- [end:function_define]
+
+# A Function version as the service returns it after registration: the SDK's
+# shared golden reply, with the name and signature of the definition above.
+SERVICE_REPLY = (
+    Path(__file__).parents[3]
+    / "rust"
+    / "lancedb"
+    / "tests"
+    / "fixtures"
+    / "first_class_functions"
+    / "v1"
+    / "remote_function_job.json"
+)
+
+
+def registered_version(definition) -> FunctionVersion:
+    request = definition.registration_request
+    reply = json.loads(SERVICE_REPLY.read_text())["result"]
+    reply["name"] = request.name
+    reply["signature"] = json.loads(request.signature.to_canonical_json())
+    return FunctionVersion.from_json(json.dumps(reply))
+
+
+def require_env(var_name: str) -> str:
+    """Skip the test unless the required environment variable is provided."""
+    value = os.environ.get(var_name)
+    if not value:
+        pytest.skip(f"Set {var_name} to run this Enterprise example")
+    return value
+
+
+def test_function_runs_locally_as_python():
+    # --8<-- [start:function_local_call]
+    print(double(1.5))
+    # 3.0
+    # --8<-- [end:function_local_call]
+    assert double(1.5) == 3.0
+
+
+def test_function_registration_request():
+    request = double.registration_request
+    assert request.name == "double"
+    assert [
+        (parameter.name, parameter.arrow_type, parameter.nullable)
+        for parameter in request.signature.inputs
+    ] == [("value", "float64", False)]
+    output = request.signature.output
+    assert (output.kind, output.arrow_type, output.nullable) == (
+        "scalar",
+        "float64",
+        False,
+    )
+    version = request.runtime.python_version
+    assert version == f"{sys.version_info.major}.{sys.version_info.minor}"
+    assert request.runtime.environment.packages == ()
+
+
+def test_function_registration_is_remote(tmp_path):
+    db = lancedb.connect(tmp_path)
+    with pytest.raises(NotImplementedError):
+        db.create_function(double)
+
+
+def test_function_version_binds_a_column(tmp_path):
+    version = registered_version(double)
+
+    application = version(value=col("value"))
+
+    assert isinstance(application, FunctionApplication)
+    assert application.function.name == "double"
+    assert application.function.version == version.version
+    assert [
+        (binding.parameter, binding.kind, binding.value["path"])
+        for binding in application.inputs
+    ] == [("value", "column", "value")]
+    with pytest.raises(TypeError, match="direct col"):
+        version(value=1.5)
+    with pytest.raises(TypeError, match="missing"):
+        version(x=col("value"))
+
+    # The binding is accepted by add_columns, which a local table refuses:
+    # a Function column needs LanceDB Cloud or Enterprise.
+    table = lancedb.connect(tmp_path).create_table("t", [{"value": 1.0}])
+    with pytest.raises(NotImplementedError, match="Cloud and Enterprise"):
+        table.add_columns({"doubled": application})
+
+
+def test_function_enterprise():
+    require_env("LANCEDB_URI")
+    require_env("LANCEDB_API_KEY")
+    require_env("LANCEDB_HOST_OVERRIDE")
+
+    # --8<-- [start:function_connect]
+    import os
+
+    import lancedb
+
+    db = lancedb.connect(
+        uri=os.environ["LANCEDB_URI"],  # db://your-database
+        api_key=os.environ["LANCEDB_API_KEY"],
+        region=os.environ.get("LANCEDB_REGION", "us-east-1"),
+        host_override=os.environ["LANCEDB_HOST_OVERRIDE"],
+    )
+    # --8<-- [end:function_connect]
+
+    table = None
+    version = None
+    try:
+        # --8<-- [start:function_create_table]
+        import pyarrow as pa
+
+        schema = pa.schema([pa.field("value", pa.float64(), nullable=False)])
+        table = db.create_table(
+            "function_demo",
+            [{"value": 1.0}, {"value": 2.0}, {"value": 3.0}],
+            schema=schema,
+            mode="overwrite",
+        )
+        # --8<-- [end:function_create_table]
+
+        # --8<-- [start:function_register]
+        job = db.create_function_async(double)
+        version = job.wait()
+        # --8<-- [end:function_register]
+        assert version.name == "double"
+
+        # --8<-- [start:function_apply]
+        from lancedb import col
+
+        table.add_columns({"doubled": version(value=col("value"))})
+        # --8<-- [end:function_apply]
+
+        # --8<-- [start:function_refresh]
+        refresh = table.refresh_column_async("doubled")
+        result = refresh.wait()
+        print(result.rows_assigned)
+        # 3
+        # --8<-- [end:function_refresh]
+        assert result.rows_assigned == 3
+        assert result.rows_failed == 0
+
+        # --8<-- [start:function_inspect]
+        rows = table.search().select(["value", "doubled"]).to_list()
+        print(sorted((row["value"], row["doubled"]) for row in rows))
+        # [(1.0, 2.0), (2.0, 4.0), (3.0, 6.0)]
+        # --8<-- [end:function_inspect]
+        assert sorted((row["value"], row["doubled"]) for row in rows) == [
+            (1.0, 2.0),
+            (2.0, 4.0),
+            (3.0, 6.0),
+        ]
+    finally:
+        if table is not None:
+            db.drop_table("function_demo")
+        if version is not None:
+            db.drop_function(version.name, version=version.version)