The phrase "a direct responsibility" is correct and usable in written English.
It can be used when discussing a specific obligation or duty that is clearly defined and assigned to someone.
Example: "As the project manager, ensuring the team's success is a direct responsibility of mine."
Alternatives: "an explicit duty" or "a clear obligation".
"Cameron has a direct responsibility for what is happening to living standards.
As in other areas the developed world has a direct responsibility to help developing countries in an extensive way.
You are benefitting from it, so you have a direct responsibility to figure out how to undo it.
These companies now have a direct responsibility to check on human rights abuses anywhere in their supply chains.
It is resulting in appalling violations of the rights of desperately vulnerable people for which the Italian authorities bear a direct responsibility and Europe's leaders a political one".
"But where we have a direct responsibility, for the stadiums for example, then we should have standards in place to make sure this doesn't happen again".
A change is confirmed in the traditional structure of the family and a more direct responsibility of mothers in taking care of their children.
If you own a manual-transmission performance car and you abuse the clutch and transmission to the point of failure, you as the owner have a pretty direct responsibility for what happened.
That's a charge of direct responsibility, and that's overstating things.
On March 29th, a congressional investigative committee published a report absolving Lula of direct responsibility for the mensalão a bribery scam that has implicated a score of congressmen, the PT's top officials and the president's former chief of staff.
A national coordinator had direct responsibility for field operations.
