Question: The answer at first sight seems quite obvious and negative.

Consider this:  

There is an electron. Right of it is a positive charge. It gets accelerated towards right. Now, instantaneously, I remove the charge on right and put a charge on the left. The acceleration changes from right to left instantly.
Won't it make an inverted 'V' like velocity time graph ? 

Now if such a graph exists then acceleration at the kink would be undefined and hence force applied at that time would be undefined which it is not at any point of time. 

PS:The distance between the electron and the positive charge is always same, though electron moves towards one charge, the charge is also moved away with some external force.

Answer: 
What you describe is a mathematically contrived situation where the forces on a particle change instantaneously. In such situations, I fall back on really boring and obvious tautologies: "If the forces on a particle change instantaneously, the forces on that particle change instantaneously." You will define the forces on this particle in a piece-wise manner. It is up to you and your mathematical games as to whether the force is defined to be positive at that instant, negative at that instant, or undefined. If it's undefined, then you can no longer integrate acceleration to get velocity, and all of your physics breaks. However, if you choose to define the velocity to be positive or negative at that moment, you can at least integrate acceleration to get velocity. You will get the "V' curve you refer to.

However, in reality, forces do not pop into existence instantaneously as you describe. Particles move continuously. You will never come across a case where you get undefined values like this. At least, you wont come across them until you get into modeling black holes. But they're a special sort of problem.
Note that the velocity time graph in this case has a sharp kink at that point implying that the function is not differentiable at that points. In any realistic situation, the functions will be differentiable at all points and the graphs will be smooth.
What this means physically is that and velocity cannot change values abruptly at an instant. Changes are always continuous.