Can I find the acceleration or velocity when my displacement-time graph is discontinuous?

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.

@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.

You can extrapolate average speed from a continuous regions separated by discontinuities.
Take a look at such function jump discontinuity case :

enter image description here

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}