# Continuity

The concept that relates functions with convergent sequences.

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Concept | Content |
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Maps | Maps are the mathematical formulation of a machine that gets inputs and generate outputs. On both sides, sets are needed for the domain and the codomain. |

Convergence | Convergent sequences have a well-defined limit. |

Limits of Functions | How function evaluations change when the argument approaches a certain point. |

__Show consequences__

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Concept | Content |
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Logarithm Function | The inverse of the exponential function. |

Differentiability | How to quantify the rate of change of a function. |

Exponential Function | A special function that can be defined via a power series. |

Continuity for Sums, Products, Quotients, and Compositions | How combination of continuous functions leads to new continuous functions. |

Continuous Images of Compact Sets Are Compact | A mapping property for continuous functions. |

Intermediate Value Theorem | This theorem tells us that continuous functions don't jump. They have to attain every value between two values in their image. |

Uniform Limits of Continuous Functions | How to preserve continuity in the limit of a sequence of continuous functions. |

Epsilon-Delta Definition | A different notion of continuity using open intervals. If the input to a continuous function varies less than delta, then the output values should vary less than epsilon. |

## Study *Continuity* #

**Definition 1** (Continuity). *Let \(I\subset\mathbb{K}\) and let \(f:I\to\mathbb{K}\) be a function. Then
\(f\) is called continuous in \(x_0\in I\) if \[\lim_{x\to x_0}f(x)=f(x_0).\] Moreover,
\(f\) is called continuous on \(I\) if it is continuous in \(x_0\) for all \(x_0\in I\).*

**Remark 2**. Sometimes we will just say \(f:I\to\mathbb{K}\) is continuous whereby we
mean it is continuous on \(I\).

**Example 3**.

The

*constant function*\(f:\mathbb{R}\to\mathbb{R}\) with \(f(x)=c\) for some \(c\in\mathbb{R}\) is continuous on \(\mathbb{R}\).The Heaviside function \[H(x)=\begin{cases}1&,\text{ if }x\geq 0,\\0&,\text{ if }x<0.\end{cases}\] is discontinuous at \(x_0=0\), but continuous everywhere else.

The function \(f:\mathbb{R}\to\mathbb{R}\) \[f(x)=\begin{cases}1&,\text{ if }x= 0,\\0&,\text{ if }x\neq0.\end{cases}\] is discontinuous at \(x_0=0\), but continuous everywhere else.

Polynomials are continuous on \(\mathbb{R}\).

*Rational functions*\(f:I\to\mathbb{K}\) with \(f(x)=\frac{p(x)}{q(x)}\) for some polynomials \(p,q\) (\(q\) is not the zero polynomial) are defined on \(I=\{x\in\mathbb{R}\,:\,q(x)\neq0\}\) and are continuous on \(I\) (due to the formulae for convergent sequences).The

*absolute value function*\(|\cdot|:\mathbb{R}\to\mathbb{R}\), i.e., \[|x|=\begin{cases}x&,\text{ if }x\geq 0,\\-x&,\text{ if }x<0.\end{cases}\] is continuous on \(\mathbb{R}\).The function \(f:\mathbb{R}\to\mathbb{R}\) with \[f(x)=\begin{cases}1&,\text{ if }x\in\mathbb{Q},\\0&,\text{ if }x\notin\mathbb{Q}\end{cases}\] is everywhere discontinuous.

*Proof:*Let \(x_0\in\mathbb{R}\):

First case: \(x_0\in\mathbb{Q}\). Then take a sequence \((x_n)_{n\in\mathbb{N}}\) with \(\lim_{n\to \infty}x_n=x_0\) and \(x_n\in\mathbb{R}\backslash\mathbb{Q}\) (for instance, \(x_n=x_0+\frac{\sqrt{2}}{n}\)). Then \(f(x_n)=0\) for all \(n\in\mathbb{N}\) and thus \(\lim_{n\to \infty}x_n=0\neq f(x_0)=1\).

Second case: \(x_0\in\mathbb{R}\backslash\mathbb{Q}\). Then take a sequence \((x_n)_{n\in\mathbb{N}}\) with \(\lim_{n\to \infty}x_n=x_0\) and \(x_n\in\mathbb{Q}\) (this exists since every real number can be approximated by a rational number in arbitrary good precision). Then \(f(x_n)=1\) for all \(n\in\mathbb{N}\) and thus \(\lim_{n\to \infty}x_n=1\neq f(x_0)=0\).

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