Math 22 Continuity

Continuity

 Informally, a function is continuous at $x=c$ means that there is no interruption in the graph of $f$ at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c}
.

Definition of Continuity

 Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c}
 be a real number in the interval Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (a,b)}
, and let $f$ be a function whose domain contains the intervalFailed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (a,b)}
 . The function $f$ is continuous at Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle c}
 when 
 these conditions are true.
 1. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(c)}
 is defined.
 2. $\lim _{x\to c}f(x)$ exists.
 3. $\lim _{x\to c}f(x)=f(c)$
 If $f$ is continuous at every point in the interval $(a,b)$, then $f$ is continuous on the open interval $(a,b)$.

Notes

Polynomial function is continuous on the entire real number line (ex: $f(x)=2x^{2}-1$ is continuous on $(-\infty ,\infty )$) Rational Functions is continuous at every number in its domain. (ex: $f(x)={\frac {x+2}{x^{2}-1}}$ is continuous on $(-\infty ,-1)\cup (-1,1)\cup (1,\infty )$)

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