The problem of instant velocity The concept of velocity is by definition the movement divided by the interval of time between initial position and final position.

If $f(t)$ is the position of a particle at time $t$; the velocity in the interval $[t_0;t_1]$ is $\dfrac{f(t_1)-f(t_0)}{t_1-t_0}$

The problem is that in a single instant there is no movement and the time is not changed; so no velocity.

I can consider $\lim_{t_1 \to t_0} \dfrac{f(t_1)-f(t_0)}{t_1-t_0}$, but mathematically it is only the limit of average velocity function and doesn't represent velocity at instant $t_0$

What are your views about this problem ?
Your excellent question is as old as the invention of calculus. As you correctly point out, velocity makes no sense if all you know is what is happening at just that instant of time. Physicists and mathematicians take the limit of the average velocity as the very definition of instantaneous velocity. 

That turns out to be a very good definition, since it leads to physics that accurately describes the behavior of the world and mathematics that's consistent and interesting and useful. So people no longer worry about the question in the form in which you've asked it.

Edits to respond to comments. Edited again (as @Polygnome suggests) to incorporate the sense of the comments as well

@pjs36 Yes indeed thanks. The question really does go back to Zeno's paradox of the arrow. On that wikipedia page you can read


  Zeno states that for motion to occur, an object must change the
  position which it occupies. He gives an example of an arrow in flight.
  He states that in any one (duration-less) instant of time, the arrow
  is neither moving to where it is, nor to where it is not.[13] It
  cannot move to where it is not, because no time elapses for it to move
  there; it cannot move to where it is, because it is already there. In
  other words, at every instant of time there is no motion occurring. If
  everything is motionless at every instant, and time is entirely
  composed of instants, then motion is impossible.


@Max says


  In the Newtonian model of the universe, momentum/velocity is something
  that objects have at every instant of time


I didn't know that. Perhaps it's why he could develop calculus reasoning with infinitesimals without addressing the philosophical problem and without formal notion of limits. His assumptions were not universally accepted at the time. The philosopher George Berkeley argued that


  ... forces and gravity, as defined by Newton, constituted "occult
  qualities" that "expressed nothing distinctly". He held that those who
  posited "something unknown in a body of which they have no idea and
  which they call the principle of motion, are in fact simply stating
  that the principle of motion is unknown."


( https://en.wikipedia.org/wiki/George_Berkeley#Philosophy_of_physics )

@leftaroundabout I agree that momentum is a better fundamental notion than velocity (certainly for quantum mechanics, possibly for Newtonian too). I don't think it's better to start calculus there, though.

@Hurkyl notes correctly that there are new mathematical structures - germs - that capture the idea of what happens near but not at a point. But I think the idea of the germ of a function is more technical and abstract than called for by the question.
"The concept of velocity is by definition the movement divided by the interval of time between initial position and final position."

Says who?

As with any other conceptual notion, the definition is whatever we deem it to a) answer a desired question while b) not contradicting itself or other connected definitions.  For example, nobody says non-integer rational exponents are to be considered comparable to roots; that's simply a convenient discovery that syncs up nicely with the already-established (by general principles of the definition of multiplication) properties of integer exponents.  It's the transitive property dropped right in our laps. 

In this case, one might simply say that "velocity" has no inherent meaning other than that which we assign to it, and we've collectively decided to treat it as if it were "naturally" the definition we mean when we actually say "average" velocity (probably because we lack the biological ability to extract ourselves from the fourth dimension and thus escape the compulsion to measure that elapsed time as having an unavoidably non-zero amount).  For all we know, we might well already have the instruments to measure instantaneous velocity; we just lack the ability to pull ourselves outside of time and actually do so.

EDIT:  OK, since my answer has rubbed a few the wrong way, let us consider it this way: it is not true to say that in an instantaneous, non-elapsed-time sense, there is "no velocity." 

"No" implies "zero." 

The velocity is not zero in this case, it is the indeterminate form 0/0.  (Despite several of my students' beliefs to the contrary, 0 &div; 0 $\not =$ 0 automatically.) It is, at best, unclear; it may be zero, or infinitely large, or some finite non-zero amount. What limits (and, through them, L'Hospital's Rule) explores is how we can explore such singularity-esque quantities only by looking at them indirectly.  My previous comment was phrased merely to suggest that we are as likely to understand instantaneous velocity as a limited case of average velocity as we are to understand average velocity as an extension of instantaneous velocity.