The core function of the project manager is?

E. Integration

Integration is a back bone of the project management, which includes all other subsidiary processes i.e. project direction, management, monitor, control, change request, produce project deliverables.

E - Integration, I suggest Communication could have been added

(A) Quality Assurance, and

(E) Integration

PM role does not fit in easily in one or two tasks. 

The role has to be broad skill set based from managing the technical aspects, triple constraints, technical and nontechnical activities, communicate, integrate and drive the team to deliver success.  

Four Core Functions: The four core functions of project management are scope, time, cost, and quality. ... Scope, which defines the boundaries of the projectand what is outside those boundaries. The boundaries are typically defined in terms of objectives, deliverables, and resources (e.g., time, personnel, money).

A. Quality assurance, since the project manager checks and verifies the status of the project he handles.

Quality Assurance and Client Interfacing

 Integration and manage work flow with time fram,Quality Assurance and the production team

The core reason to have a project Manager leading a Quality delivery team  is to co-ordinate and manage all the relationships and clear communication needed across the Business and varying teams that have key inputs and impacts of a project foremost. Yes there a tasks related to scheduling and controling and Risk management, but without the people relationships and constant communication regardless of the tools used, no project can succeed within budget or scope without a cohesive delivery team.