Difference between revisions of "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 $f(x)={\begin{cases}x+2&{\text{if }}-1\leq x<3\\14-x^{2}&{\text{if }}3\leq x\leq 5\end{cases}}$ Solution:
On the interval $[-1,3)$ , $f(x)=x+2$ and it is a polynomial function so it is continuous on $[-1,3)$ On the interval $[3,5]$ , $f(x)=14-x^{2}$ and it is a polynomial function so it is continuous on $[3,5]$ Finally we need to check if $f(x)$ is continuous at $x=3$ .
So, consider $\lim _{x\to 3^{-}}f(x)=\lim _{x\to 3^{-}}x+2=3+2=5$ Then, $\lim _{x\to 3^{+}}f(x)=\lim _{x\to 3^{+}}14-x^{2}=14-(3)^{2}=5$ .
Since $\lim _{x\to 3^{-}}f(x)=5=\lim _{x\to 3^{+}}f(x)$ , \lim_{x\to 3} f(x) exists.
Also notice $f(3)=14-(3)^{2}=5=\lim _{x\to 3}f(x)$ So by definition of continuity, $f(x)$ is continuous at $x=3$ .
Hence, $f(x)$ is continuous on $[-1,5]$ 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: $f(x)={\frac {x^{2}-2x-3}{x+1}}={\frac {(x-3)(x+1)}{x+1}}=x-3$ . This function $f(x)$ is y=x-3 with a hole at $x=-1$ since $x=-1$ makes $f(x)$ 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 $\lim _{x\to a^{-}}f(x)=\pm \infty$ or $\lim _{x\to a^{+}}f(x)=\pm \infty$ Jump discontinuity: The function is approaching different values depending on the direction $x$ is coming from. When this happens, we say the function has a jump discontinuity at $x=a$ . In another word, $\lim _{x\to a^{-}}f(x)\neq \lim _{x\to a^{+}}f(x)$ 