Hello!

NaN is not a literal, it's just an identifier.

In this case you can pass it as a character string literal:
INSERT INTO "TABLE"(ID, DOUBLE_COLUMN) VALUES (10, 'NaN');

In more complex cases where data type cannot be determined automatically a 
cast is needed:
CAST('NaN' AS DOUBLE PRECISION)

There are two other special values: CAST('Infinity' AS DOUBLE PRECISION) and 
CAST('-Infinity' AS DOUBLE PRECISION).

REAL and DECFLOAT data types also have these three special values in H2, 
but all other numeric data types (TINYINT, SMALLINT, INTEGER, BIGINT, and 
NUMERIC) don't support them.

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