Theorem 1.9.3. Continuity of \(x\).
\(x\) is continuous everywhere
Function |
Continuous on |
\begin{equation*}
x
\end{equation*}
|
\begin{equation*}
(-\infty,\infty)
\end{equation*}
|
\begin{equation*}
\sin(x)
\end{equation*}
|
\begin{equation*}
(-\infty,\infty)
\end{equation*}
|
\begin{equation*}
\cos(x)
\end{equation*}
|
\begin{equation*}
(-\infty,\infty)
\end{equation*}
|
\begin{equation*}
\tan(x)
\end{equation*}
|
\begin{equation*}
\ldots (-\pi/2,\pi/2) \cup (\pi/2,3\pi/2) \cup \ldots
\end{equation*}
|
\begin{equation*}
e^x
\end{equation*}
|
\begin{equation*}
(-\infty,\infty)
\end{equation*}
|
\begin{equation*}
\ln(x)
\end{equation*}
|
\begin{equation*}
(0,\infty)
\end{equation*}
|
Sum | If \(f(x),g(x)\) are continuous at a point, then \(f(x)+g(x)\) is continous at that point. |
Scalar | If \(f(x)\) is continuous at a point, then \(k f(x)\) is continous at that point for any scalar \(k\text{.}\)
|
Product | If \(f(x),g(x)\) are continuous at a point, then \(f(x)g(x)\) is continous at that point. |
Quotient | If \(f(x),g(x)\) are continuous at a point, then \(f(x)/g(x)\) is continous at that point if defined. |
Composition | If \(f(x),g(x)\) are continuous at a point, then \(f(g(x))\) is continous at that point when defined. |