Theorem 1.9.3. Continuity of \(x\).
\(x\) is continuous everywhere
Section 1.9 Known Continuous Functions
Rather than prove the continuity of every function we will demonstrate continuity of a few and note properties of continuity that will lead to a list of known continuous functions.
Subsection 1.9.1 Known Continuous Functions
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. |
Proof.
Consider \(f(x)=x\text{.}\)
\begin{align*}
\lim_{x \to a} x \amp = a.\\
f(a) \amp = a.
\end{align*}
Thus for all real numbers \(a\text{,}\) the function is defined, the limit is defined, and the two are equal. Thus it is continuous everywhere.
Theorem 1.9.4. Quotients preserve continuity.
If at a point \(x=a\) two functions are continuous, then their quotient is continuous at that point if the quotient is defined.
Proof.
Suppose \(f(x),g(x)\) are continuous at \(x=a\text{.}\) Consider the function \(f(x)/g(x)\text{.}\)
Note because \(f(x)\) and \(g(x)\) are continous at \(x=a\text{,}\) their limits are defined there and equal the function values. By the limit quotient property the limit of the quotient \(f(x)/g(x)\) is defined there. If \(g(a) \ne 0\) the value of the quotient is \(f(a)/g(a)\text{.}\) Thus the limit exists, the function exists, and they are equal.
Subsection 1.9.2 Practice
Checkpoint 1.9.5.
Explain why \(f(x)=10x^2-3x+9\) is continuous on \((-\infty,\infty)\text{.}\)
Checkpoint 1.9.6.
Where is \(g(\alpha)=\cot(\alpha)\) continuous?
