[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]