Question

the points of discontinuity and type of each (removable, jump, infinite, or none of these), and indicate whether f is left - or right - continuous.

1. $f(x)=lceil x

ceil$

1. $f(x)=lceil x

ceil-lfloor x
floor$
in exercises 57 - 60, sketch the graph of f. at each point of discontinuity, state whether f is left - or right - continuous.

1. $f(x)=\begin{cases}x^{2}&\text{for }xleq1\\2 - x&\text{for }x > 1end{cases}$
2. $f(x)=\begin{cases}x + 1&\text{for }x < 1\\frac{1}{x}&\text{for }xgeq1end{cases}$
3. $f(x)=\begin{cases}\frac{x^{2}-3x + 2}{vert x - 2vert}&x

eq2\\0&x = 2end{cases}$

1. $f(x)=\begin{cases}x^{3}+1&\text{for }-infty
2. show that the function $f(x)=\begin{cases}\frac{x^{2}-16}{x - 4}&x

eq4\\10&x = 4end{cases}$

Explanation:

Step1: Identify function type

$f(x)=\lfloor x
floor$ (floor function, greatest integer ≤x).

Step2: Discontinuity points

Discontinuities at all integers $n \in \mathbb{Z}$.

Step3: Limit analysis at integers

For integer $n$: $\lim_{x \to n^-} \lfloor x
floor = n-1$, $\lim_{x \to n^+} \lfloor x
floor = n$.

Step4: Discontinuity type and continuity

$LHL
eq RHL$ (jump discontinuity). $f(n)=n=RHL$, so right-continuous.

Step1: Define functions

$f(x)=\lfloor x
floor - \lceil x
ceil$ (floor - ceiling).

Step2: Evaluate for non-integers

For $x
otin \mathbb{Z}$: $\lfloor x
floor = n$, $\lceil x
ceil = n+1$, $f(x)=-1$.

Step3: Evaluate for integers

For $x = n \in \mathbb{Z}$: $\lfloor x
floor = \lceil x
ceil = n$, $f(x)=0$.

Step4: Limit and discontinuity

$\lim_{x \to n} f(x) = -1
eq f(n)=0$ (removable discontinuity at integers).

Step1: Check continuity at x=1

$f(x)=

$$\begin{cases}x^2 & x \leq1 \\ 2-x & x>1\end{cases}$$

$.

Step2: Left limit at x=1

$\lim_{x \to 1^-} x^2 = 1^2=1$.

Step3: Right limit at x=1

$\lim_{x \to 1^+} (2-x)=2-1=1$.

Step4: Function value at x=1

$f(1)=1^2=1$. $LHL=RHL=f(1)$, continuous.

Step1: Check continuity at x=1

$f(x)=

$$\begin{cases}x+1 & x<1 \\ 1/x & x \geq1\end{cases}$$

$.

Step2: Left limit at x=1

$\lim_{x \to 1^-} (x+1)=1+1=2$.

Step3: Right limit at x=1

$\lim_{x \to 1^+} (1/x)=1/1=1$.

Step4: Discontinuity type and continuity

$LHL
eq RHL$ (jump discontinuity). $f(1)=1=RHL$, right-continuous.

Answer:

Discontinuities at all integers; type: jump; right-continuous at integers.