Pointwise Properties of Real Functions of a Single Real Variable

Snapshot 1: the function is differentiable, and not tendable at .

Snapshot 2: the function is null disdetachable, and not continuous at .

Snapshot 3: the function is detachable, and not continuous at .

Continuity, differentiability, and tendability are three fundamental properties of single-variable functions. While continuity and differentiability are dependent events (if a function is differentiable then it is continuous), a function may be tendable at a point whether it is differentiable, continuous, or discontinuous there. Tendable functions are classified into three types: detachable, signposted detachable, and null disdetachable, all of which are related to the one-sided limits ; a function is said to be detachable if , signposted detachable if , and null disdetachable if .

One of the advantages of the detachment operator is that it may be defined also in cases where a function is not differentiable and even not continuous, as shown in this Demonstration.

The terms detachment and tendency are defined in a previous Demonstration. The theory of semidiscrete calculus is given in A. Finkelstein, "The Theory behind the 'Summed Area Table' Algorithm: a Simple Approach to Calculus," arXiv, 2010.