Can I find the acceleration or velocity when my displacement-time graph is discontinuous? Today, I encountered the problem where I was asked to find the velocity and acceleration from displacement-time graph but the displacement-time graph was discontinuous. So I am unable to find the velocity where it was discontinuous.
I know from my Calculus class that we cannot find the derivative when function is discontinuous at some point.
So I think it's not possible to find the velocity and acceleration at that point.
So what is the physical significance of that? 
 A: @Fakemod has answered your question perfectly. But I'd like to add an extra point. You should also know that you cannot apply calculus if the graph is non differentiable at a point, not necessarily discontinuous.
In such cases, the simple explanation is that the equations describing the motion have changed, hence, it won't make sense to use a direct integral.
A: You can extrapolate average speed from a continuous regions separated by discontinuities. Take a look at such function jump discontinuity  case :

In this case body average speed would be :
\begin{align}
v &= \frac {dx(t)}{dt} 
\\&= \frac 12 \left( \sin(t)^\prime + (2t)^\prime \right)
\\&=\frac{\cos(t)+2}{2}
\end{align}
A: In the real world, a displacement time graph can never be discontinuous. The only not-so-physical meaning that it has is that the body teleported from one place to another such that its displacement changed instantaneously/discontinuously. And as you can see, it's impossible to define the velocity of this teleportation.
A: Numerically assume then explicitly state a '$dT$' that equates to the smallest duration of time that you can ascertain from the graph/data (perhaps does your graph have a grid? If so, use the duration of a single step in time). Then calculate the difference between the discontinuous points and divide by $dT$.
Just be sure to explain it, even if your guess for $dT$ is silly you should get points for making the best of an undefined situation 
