Difference between revisions of "Math 22 Continuity"

Continuity

 Informally, a function is continuous at ${\displaystyle x=c}$ means that there is no interruption in the graph of ${\displaystyle f}$ at ${\displaystyle c}$.


Definition of Continuity

 Let ${\displaystyle c}$ be a real number in the interval ${\displaystyle (a,b)}$, and let ${\displaystyle f}$ be a function whose domain contains the interval${\displaystyle (a,b)}$ . The function ${\displaystyle f}$ is continuous at ${\displaystyle c}$ when
these conditions are true.
1. ${\displaystyle f(c)}$ is defined.
2. ${\displaystyle \lim _{x\to c}f(x)}$ exists.
3. ${\displaystyle \lim _{x\to c}f(x)=f(c)}$
If ${\displaystyle f}$ is continuous at every point in the interval ${\displaystyle (a,b)}$, then ${\displaystyle f}$ is continuous on the open interval ${\displaystyle (a,b)}$.


Notes

Polynomial function is continuous on the entire real number line (ex: ${\displaystyle f(x)=2x^{2}-1}$ is continuous on ${\displaystyle (-\infty ,\infty )}$)

Rational Functions is continuous at every number in its domain. (ex: ${\displaystyle f(x)={\frac {x+2}{x^{2}-1}}}$ is continuous on ${\displaystyle (-\infty ,-1)\cup (-1,1)\cup (1,\infty )}$ since the denominator cannot equal to zero)