Definition: Continuous Functions at Single Real Numbers

Let \(D\) be a subset of real numbers \(\mathbb R\), \(a\in D\), and let \(f:D\to\mathbb R\) be a function \(f\) is continuous at the real number \(a\in\mathbb R\), if the limit $\lim_{x\to a} f(x)=f(a)$ exists and is unique. This means that for every real sequence $(\xi_n)_{n\in\mathbb N}$ with $\xi_n\in D$ for all $n\in\mathbb N$, which convergent to $a$ we have that $\lim_{n\to\infty}f(\xi_n)=a$.

Special Case in Metric Spaces

This definition is a special case of continuous functions in metric spaces, because for any two real numbers $x,y$, the distance $|x-y|$ makes the real numbers a metric space.

Equivalent Definition

Continuous functions can also be defined using the \(\epsilon\)-\(\delta\) definition of continuity, which is proven to be equivalent to the definition of continuous functions in metric spaces. According to the \(\epsilon\)-\(\delta\) definition, \(f\) is continuous at the point \(a\), if and only if for every \(\epsilon > 0\) there is a \(\delta > 0\) such that \[|f(x)-f(a)| < \epsilon\] for all \(x\in D\) with \[|x-a| < \delta.\]

In the above definition, the general distance for metric spaces between the points $x$ and $a$ or $f(x)$ and $f(a)$ has been replaced by the absolute value of the differences $|x-a|$ and $|f(x)-f(a)|.$

Examples: 1 2 Corollaries: 1 2

  1. Proposition: Compositions of Continuous Functions on a Whole Domain
  2. Proposition: Composition of Continuous Functions at a Single Point
  3. Lemma: Functions Continuous at a Point and Non-Zero at this Point are Non-Zero in a Neighborhood of This Point
  4. Theorem: Intermediate Value Theorem
  5. Theorem: Intermediate Root Value Theorem

Corollaries: 1 2 3
Definitions: 4 5 6 7
Examples: 8 9 10 11 12 13
Explanations: 14
Lemmas: 15 16
Parts: 17 18
Proofs: 19 20 21 22 23 24 25 26 27
Propositions: 28 29 30 31 32 33 34 35
Theorems: 36

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983