How do I create a partition using a bit field in SQL Server 2014
To get partition elimination, your queries need to be explicit about the bit data type, for example:

...WHERE Archivable = CONVERT(bit, 0)
-- NOT
...WHERE Archivable = 0


Without this, the risk of truncation in the implicit conversion means partition elimination is not applied.

The implicit conversion in the CREATE PARTITION FUNCTION statement is not a contributing factor, because SQL Server will do the conversion to bit for you, once, when the function is created. Still, it would be nicer/better practice to write:

CREATE PARTITION FUNCTION [Archived](bit) 
AS RANGE LEFT 
FOR VALUES
(
    CONVERT(bit, 0), 
    CONVERT(bit, 1)
);


It might be easier all round to avoid these potential issues by making the Archived column an integer instead, with a CHECK constraint to enforce the allowed values.