# Difference between revisions of "Math 22 Continuity"

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==Continuity== | ==Continuity== | ||

− | + | ||

+ | Informally, a function is continuous at <math>x=c</math> means that there is no interruption in the graph of <math>f</math> at <math>c</math>. | ||

+ | |||

+ | ==Definition of Continuity== | ||

+ | Let <math>c</math> be a real number in the interval <math>(a,b)</math>, and let <math>f</math> be a function whose domain contains the interval <math>(a,b)</math> . The function <math>f</math> is continuous at <math>c</math> when | ||

+ | these conditions are true. | ||

+ | 1. <math>f(c)</math> is defined. | ||

+ | 2. <math>\lim_{x\to c} f(x)</math> exists. | ||

+ | 3. <math>\lim_{x\to c} f(x)=f(c)</math> | ||

+ | If <math>f</math> is continuous at every point in the interval <math>(a,b)</math>, then <math>f</math> is continuous on the '''open interval''' <math>(a,b)</math>. | ||

+ | |||

+ | ==Continuity of piece-wise functions== | ||

+ | Discuss the continuity of <math>f(x)=\begin{cases} | ||

+ | x+2 & \text{if } -1\le x<3\\ | ||

+ | 14-x^2 & \text{if } 3\le x \le 5 | ||

+ | \end{cases}</math> | ||

+ | |||

+ | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||

+ | !Solution: | ||

+ | |- | ||

+ | |On the interval <math>[-1,3)</math>, <math>f(x)=x+2</math> and it is a polynomial function so it is continuous on <math>[-1,3)</math> | ||

+ | |- | ||

+ | |On the interval <math>[3,5]</math>, <math>f(x)=14-x^2</math> and it is a polynomial function so it is continuous on <math>[3,5]</math> | ||

+ | |- | ||

+ | |Finally we need to check if <math>f(x)</math> is continuous at <math>x=3</math>. | ||

+ | |- | ||

+ | |So, consider <math>\lim_{x\to 3^-} f(x)= \lim_{x\to 3^-} x+2= 3+2=5</math> | ||

+ | |- | ||

+ | |Then, <math>\lim_{x\to 3^+} f(x)= \lim_{x\to 3^+} 14-x^2=14-(3)^2=5</math>. | ||

+ | |- | ||

+ | |Since <math>\lim_{x\to 3^-} f(x)= 5 = \lim_{x\to 3^+} f(x)</math>, \lim_{x\to 3} f(x) exists. | ||

+ | |- | ||

+ | |Also notice <math>f(3)=14-(3)^2=5=\lim_{x\to 3} f(x)</math> | ||

+ | |- | ||

+ | |So by definition of continuity, <math>f(x)</math> is continuous at <math>x=3</math>. | ||

+ | |- | ||

+ | |Hence, <math>f(x)</math> is continuous on <math>[-1,5]</math> | ||

+ | |} | ||

+ | |||

+ | ==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: <math>f(x)=\frac{x^2-2x-3}{x+1}=\frac{(x-3)(x+1)}{x+1}=x-3</math>. This function <math>f(x)</math> is y=x-3 with a hole at <math>x=-1</math> since <math>x=-1</math> makes <math>f(x)</math> 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 <math>\lim_{x\to a^-}f(x)=\pm\infty</math> or <math>\lim_{x\to a^+}f(x)=\pm\infty</math> | ||

+ | |||

+ | '''Jump discontinuity''': The function is approaching different values depending on the direction <math>x</math> is coming from. When this happens, we say the function has a jump discontinuity at <math>x=a</math>. In another word, <math>\lim_{x\to a^-}f(x)\ne\lim_{x\to a^+}f(x)</math> | ||

+ | |||

+ | ==Notes== | ||

+ | |||

+ | ''Polynomial function'' is continuous on the entire real number line (ex: <math>f(x)=2x^2-1</math> is continuous on <math>(-\infty,\infty)</math>) | ||

+ | |||

+ | ''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]]''' |

## Latest revision as of 05:51, 23 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: |
---|

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**