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Chapter 1: Limits
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Section 1.6: Continuity
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Introduction


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In the years after Newton and Leibniz promulgated the calculus, a rigorous definition of the limit was evolving. It took nearly two centuries. During this time, the notion of "continuity" was also being articulated as the analytic property of a function that reflected any "smoothness" in its graph. For example, at one time it was naively thought that a continuous function was one whose graph could be drawn without taking pencil from paper. However, $f\left(x\right)\={x}^{2}\mathrm{sin}\left(1\/x\right)$ can't be drawn through the point $x\=0$ because of the infinite oscillations, but it turns out to be "continuous." The essence of this section is a rigorous concept of "continuity" at a point, and on an interval.
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Essentials



Continuity at a Point


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A function $f$ is continuous at $x\=a$ is small changes in $x$ in the vicinity of $a$ result in small changes in the values of $f$. In other words, small changes imply small changes. This is the notion that the formal definition below captures in mathematical language.
Definition 1.6.1

The function $f\left(x\right)$ is said to be continuous at the point $x\=a$ if, for every $\mathrm{\ϵ}\>0$ there is a $\mathrm{\δ}\left(\mathrm{\ϵ}\right)$ for which $\xa\<\mathrm{\δ}\left(\mathrm{\ϵ}\right)$ ⇒ $\leftf\left(x\right)f\left(a\right)\right<\mathrm{\ϵ}$.



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Definition 1.6.1 formalizes the notion that near $x\=a$, the values of $f\left(x\right)$ stray only a little from $f\left(a\right)$ as long as the values of $x$ stray only a little from $a$. In other words, small changes in $x$ induce only small changes in $f$.
Implicit in Definition 1.6.1 is the existence of the number $f\left(a\right)$. In other words, $f$ must be defined at $x\=a$. Moreover, Definition 1.6.1 has more than a passing similarity to Definition 1.2.1, the formal definition of a limit. Indeed, Definition 1.6.1 could be rephrased as "the limit at $x\=a$ is the function value at $x\=a$." In other words, if $f$ is defined at $x\=a$, and $\underset{x\to a}{lim}f\left(x\right)\=f\left(a\right)$, then $f$ is continuous at $x\=a$. Consequently, the test for continuity amounts to checking for the three items in Table 1.6.1.
1.

The function is defined at $x\=a$ so that $f\left(a\right)$ is a real number.

2.

The limit at $x\=a$ exists.
This might mean checking that the limits from the left and right both exist and are equal.

3.

The limit at $x\=a$ equals $f\left(a\right)$, that is, $\underset{x\to a}{lim}f\left(x\right)\=f\left(a\right)$.


Table 1.6.1 Practical test for continuity at a point



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Properly understood, (3) in Table 1.6.1 is all that needs to be remembered. If it is assumed that all the symbols in $\underset{x\to a}{lim}f\left(x\right)\=f\left(a\right)$ are defined (that is, the quantities exist), then this equality captures the essence of Definition 1.6.1.
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Continuity at an Endpoint


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If $x\=a$ is the endpoint of a closed interval that is the domain of a function $f$, the question of continuity at $a$ is resolved by considering the appropriate onesided limit at $x\=a$. Thus, if $a$ is a left endpoint, then $f$ is continuous at $x\=a$ if $\underset{x\to a\+}{lim}f\left(x\right)\=f\left(a\right)$, whereas if $a$ is a right endpoint, then $f$ is continuous at $x\=a$ if $\underset{x\to a}{lim}f\left(x\right)\=f\left(a\right)$.
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Discontinuity at a Point


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A function that is not continuous at a point is said to be discontinuous at that point. A function can be discontinuous at a point in one of the four ways listed in Table 1.6.2.
Discontinuity

Example

Visualization

Limit exists, but does not equal $f\left(a\right)$ or $f\left(a\right)$ not defined

${f}_{1}\=\frac{{x}^{2}1}{x1}$


Limit does not exist because of infinite oscillation

${f}_{2}\=\mathrm{sin}\left(1\/x\right)$


Limit does not exist because onesided limits don't agree

${f}_{3}\=\left\{\begin{array}{cc}2xplus;1& x2\\ 103x& x\ge 2\end{array}\right.$


Limit does not exist because of vertical asymptote

${f}_{4}\=1\/x$


Table 1.6.2 Ways for a function to be discontinuous at a point



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•

Function ${f}_{1}$ in Table 1.6.2, undefined at $x\=1$, is said to have a removable discontinuity at $x\=1$ because $f$ can be extended to a function $g$ whose domain includes $x\=1$. Indeed, the "hole" at $\left(1\,2\right)$ does not exist for $g\left(x\right)\=x\+1$, a function that agrees with ${f}_{1}$ everywhere that ${f}_{1}$ is defined, and extends the domain of ${f}_{1}$ to include the one missing value, namely, $x\=1$.

•

The function ${f}_{3}$ in Table 1.6.2 is said to have a jump at $x\=2$ because, although the onesided limits exist, they are not equal.

•

The function ${f}_{4}$ in Table 1.6.2 is discontinuous at $x\=0$ because the onesided limits at that point don't exist. Remember, infinite limits are not real numbers, and hence, do not exist. When they occur for a finite value of the independent variable, the function has a vertical asymptote at that point. The vertical axis can be "odd," like the $y$axis in the graph of $1\/x$, or "even," like the $y$axis in the graph of $1\/{x}^{2}$.

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Continuity on an Interval


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Definition 1.6.2

•

A function that is continuous at every point in an interval I is said to be continuous on the interval I.

•

The interval I can be open, closed or halfopen.




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Continuity is a "point property" that is extended to an interval if every point in the interval has that property.
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It is impossible to test for continuity at every point in an interval. Hence, it is useful to know how primitive continuous functions combine to form (or not form) other continuous functions. To this end, see Table 1.6.3. However, before scanning the table, it is useful to sharpen intuition with a few observations:
•

The sum of continuous functions is continuous but their quotient might not be if the divisor has a zero.

•

The sum of the rules for two discontinuous functions could result in a rule that is continuous!

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This second observation needs some explaining. For example, if $f\left(x\right)\=x\+1\/x$ and $g\left(x\right)\={x}^{2}1\/x$, then both functions are discontinuous at $x\=0$, but the sum, $x\+{x}^{2}$, is the rule for a function that is continuous at $x\=0$. Looked at more critically, note that the domains of both $f$ and $g$ do not contain $x\=0$, so the domain of their sum should not contain it either (a function is a rule and a domain), and the sum is not continuous at $x\=0$. But under the "convention" that the domain of a rule is the largest set for which it is defined, the domain of $x\+{x}^{2}$ would be all the reals, and the sum would be continuous at $x\=0$.
It would seem that the contents of Table 1.6.3 shouldn't be taken lightly.
Name

Formula

Interval of Continuity

Constant

$F\left(x\right)\=c$

all real numbers: $\left(\infty \,\infty \right)$

Identity

$F\left(x\right)\=x$

all real numbers: $\left(\infty \,\infty \right)$

Absolute Value

$F\left(x\right)\=\leftx\right$

all real numbers: $\left(\infty \,\infty \right)$

Sine and Cosine

$F\left(x\right)\=\mathrm{sin}\left(x\right)$ or $G\left(x\right)\=\mathrm{cos}\left(x\right)$

all real numbers: $\left(\infty \,\infty \right)$

Tangent and Secant

$F\left(x\right)\=\mathrm{tan}\left(x\right)$ or $G\left(x\right)\=\mathrm{sec}\left(x\right)$

all real numbers except odd multiples of $\mathrm{\π}\/2$

Cotangent and Cosecant

$F\left(x\right)\=\mathrm{cot}\left(x\right)$ or $G\left(x\right)\=\mathrm{csc}\left(x\right)$

all real numbers except integer multiples of $\mathrm{\π}$

Constant Multiple

$F\left(x\right)\=cf\left(x\right)$

${I}_{1}$

Sum

$F\left(x\right)\=f\left(x\right)\+g\left(x\right)$

${I}_{1}\bigcap {I}_{2}$

Difference

$F\left(x\right)\=f\left(x\right)g\left(x\right)$

${I}_{1}\bigcap {I}_{2}$

Product

$F\left(x\right)\=f\left(x\right)g\left(x\right)$

${I}_{1}\bigcap {I}_{2}$

Quotient

$F\left(x\right)\=f\left(x\right)\/g\left(x\right)$

${I}_{1}\bigcap {I}_{2}\setminus \left\{x\:g\left(x\right)\=0\right\}\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\phantom{\rule[0.0ex]{0.0em}{0.0ex}}\=\left({I}_{1}\bigcap {I}_{2}\right)\bigcap \left\{x\:g\left(x\right)\ne 0\right\}$

Power

$F\left(x\right)\=f{\left(x\right)}^{n}$

${I}_{1}$

Root

$F\left(x\right)\=f{\left(x\right)}^{1\/n}$

$\{\begin{array}{cc}{\mathrm{I}}_{1}& n\mathrm{odd}\\ {I}_{1}\bigcap \left\{xcolon;f\left(x\right)\ge 0\right\}& n\mathrm{even}\end{array}$

Composition

$F\left(x\right)\=f\left(g\left(x\right)\right)$

$\left\{x\:g\mathrm{continuous}\mathrm{at}xandf\mathrm{continuous}\mathrm{at}g\left(x\right)\right\}$

Table 1.6.3 Continuity on an interval: $f$ continuous on ${I}_{1}$, $g$ continuous on ${I}_{2}$, $c$ is a constant, $n$ is a positive integer



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Précis


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The key ideas in this section are summarized in Table 1.6.4
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Intuitively, a continuous function $f\left(x\right)$ is one for which small changes in $x$ generate only small changes in $f\left(x\right)$.

•

Practically, $f\left(x\right)$ is continuous at $x\=a$ if $\underset{x\to a}{lim}f\left(x\right)\=f\left(a\right)$, and all entities in this equality exist.

•

Continuity for $f$ is a point property. If $f$ is continuous at every point in an interval, then $f$ is continuous on the interval.

•

Table 1.6.3 summarizes the arithmetic and algebra of continuous functions.

•

Table 1.6.2 summarizes the ways a function can be discontinuous: the removable discontinuity, the jump discontinuity, the vertical asymptote, and the infinite oscillation.

•

Continuity at the closed end of a finite interval is established by considering a onesided limit.


Table 1.6.4 Summary of this section's key ideas about continuity



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Examples


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Example 1.6.1

Show that the function $f\left(x\right)\=\{\begin{array}{cc}\frac{\mathrm{sin}\left(x\right)}{x}& x\ne 0\\ 1& x\=0\end{array}$ is continuous at $x\=0$.

Example 1.6.2

Show that $f\left(x\right)\=\sqrt{10x{x}^{2}21}$ is continuous at the endpoints of its domain.

Example 1.6.3

Show that the function $f\left(x\right)\=\mathrm{sin}\left(x\right)\/\leftx\right$ is discontinuous at $x\=0$.

Example 1.6.4

Discuss the continuity of $f\left(x\right)\=1\/\sqrt{x1}$ at $x\=1$.

Example 1.6.5

Where is the function $F\left(x\right)\=\mathrm{tan}\(\frac{\left4{x}^{3\/4}\right\+4{x}^{2}2x\+3}{{x}^{2}\+1}\)$ continuous?

Example 1.6.6

If $f\left(x\right)\=\left\{\begin{array}{cc}x1& x<1\\ 5& x\ge 1\end{array}\right.$ and $g\left(x\right)\=\left\{\begin{array}{cc}2x& x1\\ 25x& x\ge 1\end{array}\right.$, show that the rule for $\left(f\+g\right)\left(x\right)\=\left\{\begin{array}{cc}3x1& x1\\ 75x& x\ge 1\end{array}\right.$ is continuous at $x\=1$ even though neither $f$ nor $g$ is continuous at $x\=1$.



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