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it will vary linearly over the length of the element. But since each element can have different material and geometric properties, the constant terms E and I in the linear polynomial can change. Also if forces and moments are added at the nodes or the element constraints are different then the constant terms a1 and a2 will change also.
If only an external force is added to a shared node the moment will be continuous, but the element will have different polynomials and there will be a change in the slope of the moment diagram The following two figures shows this situation.
The moment diagram would be as follows:
If a moment is applied at a shared node it will be added to the moment for one or both elements that share the node, depending on the boundary conditions for the particular problem.