Math 22 Continuity

Continuity

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

Definition of Continuity

 Let $c$ be a real number in the interval $(a,b)$, and let $f$ be a function whose domain contains the interval $(a,b)$ . The function $f$ is continuous at $c$ when 
 these conditions are true.
 1. $f(c)$ 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 $(a,b)$.

Continuity of piece-wise functions

Discuss the continuity of $f(x)={\begin{cases}x+2&{\text{if }}-1\leq x<3\\14-x^{2}&{\text{if }}3\leq x\leq 5\end{cases}}$

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 $(-\infty ,-1)\cup (-1,1)\cup (1,\infty )$ since the denominator cannot equal to zero)

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