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the above definition, the general distance for metric spaces between the points $x$ and $a$ or $f(x)$ and $f(a)$ has been replaced by the absolute value of the differences $|x-a|$ and $|f(x)-f(a)|.$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

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

Github: ### References

#### Bibliography

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