Pointwise Properties of Real Functions of a Single Real Variable

Drag the slider to move through the graph's points. Simultaneously, the function's properties at the visited point are updated in the diagram below using a pink disk. You can check the box "mark interesting points" to highlight some points that demonstrate interesting cases. The graph behaves like Dirichlet's function for like the function at and around the point , and like the function at and around the point .

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.