# Math 22 Continuity

## 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 |

## Types of Discontinuity

**Removable discontinuity**: If the function factors and the bottom term cancels, the discontinuity at the x-value for which the denominator was zero is removable, so the graph has a hole in it. For example: . This function is y=x-3 with a hole at since makes undefined.

**Infinite discontinuity**: An infinite discontinuity exists when one of the one-sided limits of the function is infinite and the limit does not exist. This is an infinite discontinuity. In another word, we have infinite discontinuity when either or

**Jump discontinuity**: The function is approaching different values depending on the direction is coming from. When this happens, we say the function has a jump discontinuity at . In another word,

## 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**