Why is current finite for point charges? If an electron passes through a flat plane, then there will only be a single point in its entire path which lies on  the plane,i.e the entire charge of an electron passes through in an instant (as it is a point charge), then why isn’t the current infinite at that instant and zero at all the others?
 A: WARNING: there are many paradoxes that arise by treating an electron as a classical point particle. You are best off to avoid this concept entirely. Electrons are not classical point particles, they are particles as defined by quantum electrodynamics.
If you are still reading then you have ignored the warning, like many before you. If despite the warning you insist on speaking of a classical point charge, then you are talking about classical electromagnetism where the appropriate concept is current density. The current density of a point particle is $$\vec j(t,x,y,z)=q \ \vec v(t) \ \delta(x-r_x(t),y-r_y(t),z-r_z(t))$$ where $q$ is the charge of the point charge, $\vec v$ is its velocity, $\vec r=(r_x,r_y,r_z)$ is its position, and $\delta$ is the Dirac delta distribution.
The current density of a classical point charge is thus infinite at the location of the charge and 0 elsewhere. But as warned, this is not a good model of an electron, so it is best not to call it that.
It is far better, classically, to consider charge density to be a continuum and then current density to be simply $$\vec j = \rho \vec v$$ where $\rho$ is the charge density. This will avoid the various point charge paradoxes and is a better basis when moving to QED and how electrons are actually treated.
A: Yes, single point particle with finite charge crossing a control plane means infinite current on that plane, in that instant of time. However, this infinite current does not last for any finite amount of time; it is there only for that instant, i.e. zero time interval.
If there are more such particles, we have current that is zero most of the time, and infinite at few special time instants.
Infinite current for zero time is not really a problem. If it bothers you, don't think in terms of instantaneous current, but in terms of average current, i.e. charge transported through the control plane during some chosen unit of time. This average current is usually finite.
