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Calculus I: Activities

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

Table 1.9.1. 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*}
Table 1.9.2. Algebraic Properties of Limits
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.

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?