Proving discontinuity – "Math for Non-Geeks"

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

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Epsilon-Delta Criterion

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General proof structure

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

${\color {Red}\exists \epsilon >0}\ {\color {RedOrange}\forall \delta >0}\ {\color {OliveGreen}\exists x\in D}:{\color {DarkOrchid}|x-x_{0}|<\delta }\land {\color {Blue}|f(x)-f(x_{0})|\geq \epsilon }$

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

${\begin{array}{l}{\color {Red}\underbrace {{\underset {}{}}{\text{Choose }}\epsilon =\ldots } _{\exists \epsilon >0}}\ {\color {RedOrange}\underbrace {{\underset {}{}}{\text{Let }}\delta >0{\text{ beliebig.}}} _{\forall \delta >0}}\\{\color {OliveGreen}\underbrace {{\underset {}{}}{\text{Choose }}x=\ldots {\text{ Es ist }}x\in D{\text{, weil}}\ldots } _{\exists x\in D}}\\{\color {Black}{\text{We have:}}}\\[0.5em]\quad \quad {\color {DarkOrchid}{\text{Proof for }}|x-x_{0}|<\delta }\\[0.5em]\quad \quad {\color {Blue}{\text{Proof for }}|f(x)-f(x_{0})|\geq \epsilon }\end{array}}$