I have an idea why a function f cannot be an eigenfunction of 2 different non-commutating operators, but has anyone a nicely precise reason??

If the angular momentum is resolved into its components, and we look at one, say L_z, then:

L_z.f = h.m.f

I am letting h be h-bar, m the quantum number, f the wave function, and L_z is the operator/

Is the total angular momentum quantized in m then also?? How do we find the total angular momentum from just this equation for a COMPONENT?

is it possible to select a direction so that the total angular momentum is L_z?