If an Electrical Field can jump over a point on its stright path of propagation? Consider point B between points A and C on a stright line in vaccum(or any other environment). If the electrical fild $\vec E$ (or an EM wave) should necessarily pass through B to affect C and appear there on C?
I mean if on a path of a propagating field we should always have a continuous set of points which the mentioned field affect and pass through them? Why/Why not?
 A: 
Consider point B between points A and C on a stright line in vaccum(or any other environment). If the electrical fild E⃗  (or an EM wave) 

There is a big difference between an Electric Field and an electromagnetic wave.
The electric field is a static description of the force felt by a point charge in space. It is a static picture, nothing is going anywhere when one has an electric field. There is no way of skipping anything.
The electromagnetic wave has two frameworks, the classical and the quantum mechanical. Classically there is continuity in the motion of the wave, which has a beginning and an end and is emitted by a source, and antenna, an incandescent lamp, etc.
Classically it cannot skip B.

should necessarily pass through B to affect C and appear there on C? I mean if on a path of a propagating field we should always have a continuous set of points which the mentioned field affect and pass through them? Why/Why not?

Again, for the EM wave:
Quantum mechanically the EM wave is composed by an ensemble of photons. Each individual photon is constrained by the Heisenberg Uncertainty principle and depending on the accuracy of our measurement of the momentum of the indvidual photon there is a probability within the HUP it may skip B. But as the wave is composed by an ensemble of photons and the transition between classical and quantum mechanical formats is smooth, another photon will pass B and it will be impossible to determine a skip before reaching C.
