Discrete calculus is an analog of the continuous version where the 'shift parameter' $h$ remains a non-zero positive number instead of being passed to a limit.
Discrete calculus is an analog of real calculus where the difference $h$ does not pass to a limit. For example, compare the usual derivative $f'$ and the discrete (forward-) difference $\Delta$: $$ f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\qquad \Delta f(x):= f(x+1)-f(x) $$Many continuous objects have a discrete analog. For instance, $\Delta 2^n = 2^n$, and the Bernoulli polynomials $B_n(x)$ satisfy $\Delta B_n(x) = n x^{n-1}$. Definite and indefinite integrals can be defined as well using infinite series.
Discrete calculus is used in pure and applied areas, ranging from image processing to number theory. Consider using other tags to hone the focus of your question.
