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Apex only for force.com hosted apps?
Is this use of Apex only permitted on “native” applications that are hosted on force.com? Or is Apex also available for external applications to hit the “Open APIs” such as REST API and Bulk API?
Apex is language dependent on Force.com platform,you can't run Apex other than Force.com platform. But,the Rest API webservices which you would create using Force.com platform can be accessible by any external application.
Regarding your Bulk API question,it is a feature provided by salesforce to run bulk operation so that you can perform operations on large volume of records and you don't hit your Standard APIs limit.It is not like any Open APIs like SOAP or Rest API,it's a feature of salesforce.
To learn more about Bulk API,go through below link :-
https://developer.salesforce.com/page/Bulk_API
https://developer.salesforce.com/page/Loading_Large_Data_Sets_with_the_Force.com_Bulk_API
Hope this helps !!