How is it possible to differentiate or integrate with respect to discrete time or space?

Let's say space is really a lattice with spacing $\Delta x$. It turns out that this idea has more trouble with experiment than you might think, but we can plow ahead for the purposes of this question.

You might propose replacing integrals in physics with discrete sums over individual lattice points, to take a concrete example let's think about the work needed to move a particle from point $A$ to point $B$ \begin{equation} W = \int_A^B \vec{F} \cdot {\rm d} \vec{x} \rightarrow \sum_{i=1}^N \vec{F}(\vec{x}_i) \cdot \hat{e}_{i,i+1} \Delta x \end{equation} where $i=1,2,...,N$ labels the lattice points that the particle follows in going from $A$ to $B$ and $\hat{e}_{i,i+1}$ is a vector pointing from the lattice space at $i$ to the lattice point at $i+1$.

If $\Delta x$ is small enough so $N$ is large enough, these two quantities will be quite close (since in the limit of infinite $N$ the two quantities are actually exactly the same). To see a difference (if there is one) we need to probe distances of the same order or smaller than $\Delta x$, or else have a large precision to tell the difference between these two expressions.

Here's the point. No one has ever found any disagreement between experiment and theory that can be attributable to the failure of the continuum limit. If there is such a $\Delta x$, it must be so small that it is a very good approximation to use integrals instead of sums over lattice in all experiments done to date. You can think of the LHC as probing energy scales of order 1-10 TeV, which amounts to $10^{-18}-10^{-19}$ meters -- so $\Delta x$, if it is nonzero, must be smaller than this.

There are other problems with having a lattice, but this is already a powerful argument that the world is at least effectively continuous at the scales we can probe.

This is a comment, as Andrew's answer is adequate for the problem.

I want to point out , which is not clear in your question, the difference between mathematical modeling and the object modeled.

When modeling an object mathematically one can use continuous variables by the function of mathematics. If the object modeled has discontinuities, the mathematics will model it with continuous variables. A crystal lattice for example can be modeled in continuous (x,y,z) variables with discontinuities in the mathematical model where the atoms are.

In the case of space-time there are really two concepts overlapping: the mathematical variables of space and time to be used in modeling it, and the object modeled, i.e. space and time. The mathematical variables to model a latticed space-time can be continuous, and give the functional form of the latticed physical space-time. The $x$ in Andrews answer is still a continuous variable used to model a latticed space.