Serlo: EN: Proving continuity

{{#invoke:Mathe für Nicht-Freaks/Seite|oben}}

Overview

There are several methods for proving continuity:

• Concatenation Theorems: If the function can be written as a concatenation of continuous functions, it's continuous by the Concatenation Theorems.
• Using the local nature of continuity: If a function looks like another well-known continuous function in a small neighborhood of a point, then it must also be continuous in this point.
• Considering the left- and right-sided limits: If one can show that the left- and right-sided limits of a function are the same in some point, then the function is continuous in this point.
• Showing the sequence criterion: Using the sequence criterion means that the limit can be pulled into the function, i.e. we can consider the limit of the arguments. For a sequence $(x_n{)}_{n\in \mathbb{N}}$ of arguments with limit value $x_0$ it must hold $\underset{n\to \mathrm{\infty }}{\lim }f(x_n)=f(x_0)=f\left(\underset{n\to \mathrm{\infty }}{\lim }{x}_n\right)$.
• Showing the epsilon-delta criterion: For every $ϵ>0$ it must be show that there exists some $\delta >0$ such that for all arguments $x$ with a distance smaller than $\delta$ from the point $x_0$, the inequality $|f(x)-f(x_0)|<ϵ$ is satisfied.

Composition of continuous functions

General proof sketch

{{#lst:Serlo: EN: Composition of continuous functions|Sketch of the proof}}

Example Problem

{{#lst:Serlo: EN: Composition of continuous functions|Problem:Epsilon-delta proof of continuity for a square root function}}

Using the Local Nature of Continuity

If applicable, one can use the fact that continuity is a local property. Namely, when a function $f$ looks the same as another function $g$ in a certain point $x$, i.e. $f(x)=g(x)$ on some small neighborhood of $x$, and we know $g$ to be continuous, then $f$ must also be continuous at $x$. For example, consider the function $f:ℝ\setminus \{0\}\to ℝ$ with $f(x)=1$ for positive $x$ and $f(x)=-1$ for negative $x$. Now we choose some random positive number $x_0$. In a sufficiently small neighborhood of $x_0$, $f$ is constant $1$:

The function f is constantly 1 in a sufficiently small neighborhood of a positive argument.

Since constant functions are continuous, $f$ is also continuous at the point $x_0$. Similarly one can show that $f$ is also locally constant for negative number. Then $f$ is also continuous for negative numbers, and is therefore continuous for all real numbers. In the proof we write:

Such an argument can often be applied to functions that are defined by a case differential. Our function $f$ is a good example of that. In conclusion, it's defined as:

However, the local nature of continuity can not be used as an argument for every function defined by a case differential! Let's consider the following function $g$:

For all points that are not zero we can construct a proof like we've done earlier in this chapter to show that the function is continuous at these points. However, at the point $x_0=0$ we have to construct a different argument. For example, here we could consider the left- and right-sided limits.

Sequence Criterion

Review: Sequence Criterion

{{#lst:Serlo: EN: Sequential definition of continuity|Sequence criterion of continuity}}

General Proof Structure

{{#lst:Serlo: EN: Sequential definition of continuity|Beweisskizze:Stetigkeit}}

Example Problem

{{#lst:Serlo: EN: Sequential definition of continuity|Continuity of quadratic functions}}

Epsilon-Delta Criterion Vorlage:Anker

Review: Epsilon-Delta Criterion

{{#lst:Serlo: EN: Epsilon-delta definition of continuity|Definition}}

General Proof Structure

In mathematical quantifiers, the epsilon-delta definition of continuity of a function $f$ at the point $x_0$ reads:

This technical method of writing the claim specifies the general proof structure for proving continuity using the epsilon-delta criterion:

Example Problem and General Procedure

{{#lst:Serlo: EN: Epsilon-delta definition of continuity|Problem:Quadratic function}}

Example Problems

Sequence Criterion: Absolute Value Function

{{#lst:Serlo: EN: Sequential definition of continuity|Exercise:Continuity of the absolute function}}

Epsilon-Delta Criterion: Linear Function

{{#lst:Serlo: EN: Epsilon-delta definition of continuity|Problem:Continuity of a linear function}}

Epsilon-Delta Criterion: Concatenated Absolute Value Function

{{#lst:Serlo: EN: Epsilon-delta definition of continuity|Problem:Concatenated absolute function}}

Epsilon-Delta Criterion: Hyperbola

{{#lst:Serlo: EN: Epsilon-delta definition of continuity|Exercise:Hyperbola}}

Epsilon-Delta Criterion: Concatenated Root Function

{{#lst:Serlo: EN: Composition of continuous functions|Problem:Epsilon-delta proof of continuity for a square root function}}

{{#invoke:Mathe für Nicht-Freaks/Seite|unten}}