Pointwise Properties of Real Functions of a Single Real Variable

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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 .

Contributed by: Amir Finkelstein (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

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.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send