Difference between revisions of "Math 22 Continuity"
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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== | ==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> | ||
==Notes== | ==Notes== | ||
Revision as of 08:10, 16 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 the open interval .
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}}
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)
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