Math for Non-Geeks/ Proving discontinuity

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Overview

Recall that whenever we want to prove the negation of a statement about some elements of a set, we need to show that there as at least one element in that set for which the statement is not true. So, in order the prove the discontinuity of a function, all you have to show is that the function has (at least) one point of discontinuity. There are several methods available for proving the existence of a point of discontinuity:

• Sequence criterion: Show that the function doesn't fulfill the sequence criterion at one particular point
• Considering the left- and right-sided limits: Calculate the left-sided and right-sided limits of the function at a particular point. If either one of these limits doesn't exist, or if the limits are different, then the function is discontinuous at that point.
• Epsilon-Delta Criterion: Show that the function doesn't fulfill the epsilon-delta criterion at a particular point.

Sequence Criterion

Review: Sequence Criterion

{{#lst:Math for Non-Geeks: Sequential definition of continuity|Sequence criterion of continuity for a single argument}}

Sketch of the proof

{{#lst:Math for Non-Geeks: Sequential definition of continuity|Sketch of proof:discontinuity}}

Example exercises

{{#lst:Math for Non-Geeks: Sequential definition of continuity|Exercise:Discontinuity of the topological sine function}}

Epsilon-Delta Criterion

Review: Epsilon-Delta Criterion

{{#lst:Math for Non-Geeks: Epsilon-delta definition of continuity|Definition:discontinuity}}

General proof structure

The Epsilon-Delta criterion of discontinuity can be formulated in predicate logic as follows:

From here we get a schematic that allows us the prove the discontinuity of a function using the delta-epsilon criterion:

Example exercise

{{#lst:Math for Non-Geeks: Epsilon-delta definition of continuity|Problem:Topological sine function}}

Exercises

Epsilon-delta criterion: Signum function

{{#lst:Math for Non-Geeks: Epsilon-delta definition of continuity|Problem:Discontinuity of the signum function}}

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