I can't understand why you're asking "Is distance, velocity a function of time?". The question is quite ambiguous because, when we define velocity, acceleration, or jerk in classical mechanics, we're quite sure that we're taking the time derivative of the predecessor. For instance, if you require velocity, then you're taking the time derivative of distance.

$$v(t)=\frac{dx}{dt}=\lim_{\delta t \to 0}\frac{x(t+\delta t)-x(t)}{\delta t}$$

The positions should necessarily be a function of time in order for taking the time derivative. This expression for average velocity simply means that we're putting some digits $\delta t$ to the initial state (position) of the system and determine how the system responds to it (i.e) how it moves (whether it moves or not) along the spatial axis. If it has some finite velocity, its position changes to some other value corresponding to the added time period. Finally, dividing it with the same time period which is to predict how the position is changing over time.

The expression says how the position has changed (numerator) within some period of time (denominator). If $x$ is a function of velocity, then we can say that we multiply it with $t$ and then integrate over a certain limits which you wanna predict. You're somehow arriving to the point that it is a $f(t)$.

What's my point is that units should be conserved when dealing with physical parameters. Whatever you play around (using math) with those expressions, be sure that you arrive at the final conclusion that the velocity is always $m/s$ (in SI)...

I can't understand why you're asking "Is distance, velocity a function of time?". The question is quite ambiguous because, when we define velocity, acceleration, or jerk in classical mechanics, we're quite sure that we're taking the time derivative of the predecessor. For instance, if you require velocity, then you're taking the time derivative of distance.

$$v(t)=\frac{dx}{dt}=\lim_{\delta t \to 0}\frac{x(t+\delta t)-x(t)}{\delta t}$$

The positions should necessarily be a function of time in order for taking the time derivative. This expression for average velocity simply means that we're putting some digits $\delta t$ to the initial state (position) of the system and determine how the system responds to it (i.e) how it moves (whether it moves or not) along the spatial axis. If it has some finite velocity, its position changes to some other value corresponding to the added time period. Finally, dividing it with the same time period which is to predict how the position is changing over time.

The expression says how the position has changed (numerator) within some period of time (denominator). If $x$ is a function of velocity, then we can say that we multiply it with $t$ and then integrate over a certain limits which you wanna predict. You're somehow arriving to the point that it is a $f(t)$.

What's my point is that units should be conserved when dealing with physical parameters. Whatever you play around (using math) with those expressions, be sure that you arrive at the final conclusion that the velocity is always $m/s$ (in SI)...

then its velocity must remain constant. [...] the distance... ...should remain constant [...] the difference between the velocities should remain constant

There is nothing that the particle should or must follow some trajectory or the laws that we define. We just approximate our current laws accordingly to its activity. So, the answer - It's not necessary..!