exercise:726b0fb007: Difference between revisions
From Stochiki
(Created page with "<div class="d-none"><math> \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow}...") |
No edit summary |
||
| Line 32: | Line 32: | ||
\newcommand{\mathds}{\mathbb} | \newcommand{\mathds}{\mathbb} | ||
</math></div> | </math></div> | ||
For each of the following functions, find those numbers (if any) | |||
at which the function is not continuous. | For each of the following functions, find those numbers (if any) at which the function is not continuous. | ||
<ul style{{=}}"list-style-type:lower-alpha"><li><math>x^3 + 3x - 1</math></li> | <ul style{{=}}"list-style-type:lower-alpha"><li><math>x^3 + 3x - 1</math></li> | ||
<li><math>f(x) = |x|</math></li> | <li><math>f(x) = |x|</math></li> | ||
| Line 40: | Line 41: | ||
<li><math>\sqrt{x+3}</math></li> | <li><math>\sqrt{x+3}</math></li> | ||
<li><math>h(x) = \frac{|x|}x</math></li> | <li><math>h(x) = \frac{|x|}x</math></li> | ||
<li><math>f(x) = \ | <li><math>f(x) = \begin{cases} |x|, |x| \leq 1 \\ 2-x^2, |x| > 1 \end{cases}</math></li> | ||
<li><math>F(x)= \ | <li><math>F(x)= \begin{cases}1, \mbox{if $x$ is rational} \\ 0, \mbox{if $x$ is irrational}\end{cases}</math></li> | ||
<li><math>f(x)= \ | <li><math>f(x)= \begin{cases}x^2 +2x +1, \mbox{if $x \ne 1$} | ||
{x^2 +2x +1, | \\1, \mbox{if $x = 1$}\end{cases} | ||
</math></li> | </math></li> | ||
</ul> | </ul> | ||
Latest revision as of 00:15, 23 November 2024
[math]
\newcommand{\ex}[1]{\item }
\newcommand{\sx}{\item}
\newcommand{\x}{\sx}
\newcommand{\sxlab}[1]{}
\newcommand{\xlab}{\sxlab}
\newcommand{\prov}[1] {\quad #1}
\newcommand{\provx}[1] {\quad \mbox{#1}}
\newcommand{\intext}[1]{\quad \mbox{#1} \quad}
\newcommand{\R}{\mathrm{\bf R}}
\newcommand{\Q}{\mathrm{\bf Q}}
\newcommand{\Z}{\mathrm{\bf Z}}
\newcommand{\C}{\mathrm{\bf C}}
\newcommand{\dt}{\textbf}
\newcommand{\goesto}{\rightarrow}
\newcommand{\ddxof}[1]{\frac{d #1}{d x}}
\newcommand{\ddx}{\frac{d}{dx}}
\newcommand{\ddt}{\frac{d}{dt}}
\newcommand{\dydx}{\ddxof y}
\newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}}
\newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}}
\newcommand{\dist}{\mathrm{distance}}
\newcommand{\arccot}{\mathrm{arccot\:}}
\newcommand{\arccsc}{\mathrm{arccsc\:}}
\newcommand{\arcsec}{\mathrm{arcsec\:}}
\newcommand{\arctanh}{\mathrm{arctanh\:}}
\newcommand{\arcsinh}{\mathrm{arcsinh\:}}
\newcommand{\arccosh}{\mathrm{arccosh\:}}
\newcommand{\sech}{\mathrm{sech\:}}
\newcommand{\csch}{\mathrm{csch\:}}
\newcommand{\conj}[1]{\overline{#1}}
\newcommand{\mathds}{\mathbb}
[/math]
For each of the following functions, find those numbers (if any) at which the function is not continuous.
- [math]x^3 + 3x - 1[/math]
- [math]f(x) = |x|[/math]
- [math]\frac{x^3+x+1}{x^2-x-2}[/math]
- [math]g(x) = \frac{x^3-3x-2}{x-2}[/math]
- [math]\sqrt{x+3}[/math]
- [math]h(x) = \frac{|x|}x[/math]
- [math]f(x) = \begin{cases} |x|, |x| \leq 1 \\ 2-x^2, |x| \gt 1 \end{cases}[/math]
- [math]F(x)= \begin{cases}1, \mbox{if $x$ is rational} \\ 0, \mbox{if $x$ is irrational}\end{cases}[/math]
- [math]f(x)= \begin{cases}x^2 +2x +1, \mbox{if $x \ne 1$} \\1, \mbox{if $x = 1$}\end{cases} [/math]