# Difference between revisions of "Math 22 Continuity"

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''Rational functions'' is continuous at every number in its domain. (ex: <math>f(x)=\frac {x+2}{x^2-1}</math> is continuous on <math>(-\infty,-1)\cup (-1,1)\cup (1,\infty)</math> since the denominator cannot equal to zero) | ''Rational functions'' is continuous at every number in its domain. (ex: <math>f(x)=\frac {x+2}{x^2-1}</math> is continuous on <math>(-\infty,-1)\cup (-1,1)\cup (1,\infty)</math> since the denominator cannot equal to zero) | ||

− | + | [[Math_22| '''Return to Topics Page''']] | |

'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' |

## Revision as of 06:51, 19 July 2020

## Continuity

Informally, a function is continuous at means that there is no interruption in the graph of at .

## Definition of Continuity

Let be a real number in the interval , and let be a function whose domain contains the interval . The function is continuous at when these conditions are true. 1. is defined. 2. exists. 3. If is continuous at every point in the interval , then is continuous on theopen interval.

## Continuity of piece-wise functions

Discuss the continuity of

Solution: |
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On the interval , and it is a polynomial function so it is continuous on |

On the interval , and it is a polynomial function so it is continuous on |

Finally we need to check if is continuous at . |

So, consider |

Then, . |

Since , \lim_{x\to 3} f(x) exists. |

Also notice |

So by definition of continuity, is continuous at . |

Hence, is continuous on |

## Notes

*Polynomial function* is continuous on the entire real number line (ex: is continuous on )

*Rational functions* is continuous at every number in its domain. (ex: is continuous on since the denominator cannot equal to zero)

**This page were made by Tri Phan**