An nth Riemann derivative of a function f at a point x is a limit of the form \( D_{\mathcal {A}}f(x)=\lim _{h\rightarrow 0}\sum _{i=0}^na_if(x+b_ih)/h^n, \) where the coefficients \(a_i\) and nodes \(b_i\) satisfy the Vandermonde linear system \(\sum _i a_ib_i^j=n!\cdot \delta _{nj}\) for \(j=0,1,\ldots ,n\) . The function f is n times Peano differentiable at x if it is approximated to order n near x by its nth Taylor polynomial. In 1936, Marcinkiewicz and Zygmund proved that the nth Riemann differentiability \(\widetilde{D}_n\) with nodes \(0,1,2,2^2,\ldots ,2^{n-1}\) makes up the difference between the \(n-1\) st and nth Peano differentiabilities for all functions f at x. Call each \(D_{\mathcal {A}}\) with the same property an MZ-differentiation. A number of recent results on MZ-differentiation have opened this subject of classical analysis to ideas from linear and abstract algebra, number theory, or combinatorics. This largely expository article outlines many of these results using numerous examples and counterexamples to illustrate the theorems and give insight into some of the harder proofs. The topics include first order MZ-differentiations, Gaussian differentiations, symmetric and forward Riemann differentiations, the special third Riemann differentiation with nodes \(-1,0,1,2\) , Riemann differentiations with geometric nodes, and the connection with the classification of generalized Riemann derivatives.
