  Though a core element of the digital age, numerical difference algorithms
struggle with noise susceptibility. This stems from a key disconnect between
the infinitesimal quantities in continuous differentiation and the finite
intervals in its discrete counterpart. This disconnect violates the fundamental
definition of differentiation (Leibniz and Cauchy). To bridge this gap, we
build a novel general difference (Tao General Difference, TGD). Departing from
derivative-by-integration, TGD generalizes differentiation to finite intervals
in continuous domains through three key constraints. This allows us to
calculate the general difference of a sequence in discrete domain via the
continuous step function constructed from the sequence. Two construction
methods, the rotational construction and the orthogonal construction, are
proposed to construct the operators of TGD. The construction TGD operators take
same convolution mode in calculation for continuous functions, discrete
sequences, and arrays across any dimension. Our analysis with example
operations showcases TGD's capability in both continuous and discrete domains,
paving the way for accurate and noise-resistant differentiation in the digital
era.
