Question

in exercises 17 - 38, determine the points of discontinuity. state the type of discontinuity (removable, jump, infinite, or none of these) and whether the function is left - or right - continuous.

1. $f(x)=\frac{1}{x}$
2. $f(x)=|x|$
3. $f(x)=\frac{x - 2}{|x - 1|}$
4. $f(x)=lfloor x

floor$

1. $f(x)=lfloor\frac{x}{2}

floor$

1. $g(t)=\frac{1}{t^{2}-1}$
2. $h(x)=\frac{1}{2 - |x|}$
3. $k(x)=\frac{x - 2}{|2 - x|}$
4. $f(x)=\frac{x + 1}{4x - 2}$

Explanation:

Step1: Identify domain

$f(x)=\frac{1}{x}$ is undefined at $x=0$.

Step2: Check limits

$\lim_{x \to 0^-} \frac{1}{x}=-\infty$, $\lim_{x \to 0^+} \frac{1}{x}=+\infty$.

Step1: Check definition and limits

$f(x)=|x|$ is defined everywhere. $\lim_{x \to a} |x|=|a|=f(a)$ for all $a$.

Step1: Identify domain

$f(x)=\frac{x-2}{|x-1|}$ is undefined at $x=1$.

Step2: Check limits

$\lim_{x \to 1^-} \frac{x-2}{1-x}=-\infty$, $\lim_{x \to 1^+} \frac{x-2}{x-1}=-\infty$.

Step1: Analyze floor function

$f(x)=\lfloor x
floor$ has jumps at all integers $n$.

Step2: Limits at integers

$\lim_{x \to n^-} \lfloor x
floor =n-1$, $\lim_{x \to n^+} \lfloor x
floor =n=f(n)$.

Answer:

Discontinuity at $x=0$; type: infinite; neither left nor right continuous.