Difference between revisions of "Math 22 Continuity"

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==Continuity==
 
==Continuity==
  
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  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>.
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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==
 
==Definition of Continuity==
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   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  
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   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.
 
   these conditions are true.
 
   1. <math>f(c)</math> is defined.
 
   1. <math>f(c)</math> is defined.
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   3. <math>\lim_{x\to c} f(x)=f(c)</math>
 
   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>.
 
   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}
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x+2 & \text{if } -1\le x<3\\
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14-x^2 & \text{if } 3\le x \le 5
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\end{cases}</math>
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 +
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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!Solution: &nbsp;
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|-
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|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>
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|-
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|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>
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|-
 +
|Finally we need to check if <math>f(x)</math> is continuous at <math>x=3</math>.
 +
|-
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|So, consider <math>\lim_{x\to 3^-} f(x)= \lim_{x\to 3^-} x+2= 3+2=5</math>
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|-
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|Then, <math>\lim_{x\to 3^+} f(x)= \lim_{x\to 3^+} 14-x^2=14-(3)^2=5</math>.
 +
|-
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|Since <math>\lim_{x\to 3^-} f(x)= 5 = \lim_{x\to 3^+} f(x)</math>, \lim_{x\to 3} f(x) exists.
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|-
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|Also notice <math>f(3)=14-(3)^2=5=\lim_{x\to 3} f(x)</math>
 +
|-
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|So by definition of continuity, <math>f(x)</math> is continuous at <math>x=3</math>.
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|-
 +
|Hence, <math>f(x)</math> is continuous on <math>[-1,5]</math>
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|}
 +
 +
==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==
 
==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>)
 
''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>)
 
  
 +
''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)
  
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[[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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=c} means that there is no interruption in the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

Definition of Continuity

 Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}
 be a real number in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
, and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}
 be a function whose domain contains the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
 . The function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}
 is continuous at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}
 when 
 these conditions are true.
 1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(c)}
 is defined.
 2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} f(x)}
 exists.
 3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c} f(x)=f(c)}

 If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}
 is continuous at every point in the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}
 is continuous on the open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a,b)}
.

Continuity of piece-wise functions

Discuss the continuity of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\begin{cases} x+2 & \text{if } -1\le x<3\\ 14-x^2 & \text{if } 3\le x \le 5 \end{cases}}

Solution:  
On the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-1,3)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x+2} and it is a polynomial function so it is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-1,3)}
On the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3,5]} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=14-x^2} and it is a polynomial function so it is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3,5]}
Finally we need to check if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is continuous at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=3} .
So, consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 3^-} f(x)= \lim_{x\to 3^-} x+2= 3+2=5}
Then, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 3^+} f(x)= \lim_{x\to 3^+} 14-x^2=14-(3)^2=5} .
Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to 3^-} f(x)= 5 = \lim_{x\to 3^+} f(x)} , \lim_{x\to 3} f(x) exists.
Also notice Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(3)=14-(3)^2=5=\lim_{x\to 3} f(x)}
So by definition of continuity, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is continuous at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=3} .
Hence, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [-1,5]}

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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac{x^2-2x-3}{x+1}=\frac{(x-3)(x+1)}{x+1}=x-3} . This function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is y=x-3 with a hole at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=-1} since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=-1} makes Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a^-}f(x)=\pm\infty} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a^+}f(x)=\pm\infty}

Jump discontinuity: The function is approaching different values depending on the direction Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is coming from. When this happens, we say the function has a jump discontinuity at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=a} . In another word, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a^-}f(x)\ne\lim_{x\to a^+}f(x)}

Notes

Polynomial function is continuous on the entire real number line (ex: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=2x^2-1} is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty,\infty)} )

Rational functions is continuous at every number in its domain. (ex: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac {x+2}{x^2-1}} is continuous on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty,-1)\cup (-1,1)\cup (1,\infty)} since the denominator cannot equal to zero)

Return to Topics Page

This page were made by Tri Phan

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