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\begin_body

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%MODULE: Continuity
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\begin_layout Title
Continuous functions
\end_layout

\begin_layout Subtitle
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{
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\backslash
hilite
\end_layout

\end_inset

 Continuous
\begin_inset ERT
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}
\end_layout

\end_inset

 functions possess a special property that make many investigations easy.
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{
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Defining the problem
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}
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\end_inset


\end_layout

\begin_layout Standard
\noindent
In previous modules we have seen that
\begin_inset Formula 
\begin{align*}
\ec\ec\lim_{x\rightarrow1}x^{2} & \ec=1^{2}=1.
\end{align*}

\end_inset

 Notice that you could find this answer simply by substituing 
\begin_inset Formula $\ec x=1$
\end_inset

 into the function 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

.
 Similarly, either the algebraic, graphical, or numerical approaches will
 show that
\begin_inset Formula 
\begin{align*}
\ec\lim_{x\rightarrow4}(x^{3}-4x^{2}+3) & \ec=4^{3}-4\cdot4^{2}+3=3\\
\ec\lim_{x\rightarrow0}\left|x+2\right| & \ec=\left|0+2\right|=2.
\end{align*}

\end_inset

Many other limits can be found simply by substituting the 
\begin_inset Formula $x$
\end_inset

 value.
\end_layout

\begin_layout Standard
On the other hand, many limits 
\emph on
cannot
\emph default
 be found simply by substituting the 
\begin_inset Formula $x$
\end_inset

 value.
 You have seen this with
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0}\sin\left(\frac{1}{x}\right)}\qquad\mathrm{or}{\ec\qquad\lim_{x\rightarrow1}\frac{x^{2}-x}{x-1}}.
\]

\end_inset


\end_layout

\begin_layout Standard
What we would like to know is, 
\emph on
which
\emph default
 functions enjoy the property that we can find the limit simply by susbstituting
 the 
\begin_inset Formula $x$
\end_inset

 value?
\end_layout

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%2
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{
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Redefining the problem
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\begin_layout Standard
Let's consider the question carefully:
\end_layout

\begin_layout Standard
\noindent
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\backslash
begin{question}
\end_layout

\end_inset


\emph on
Which
\emph default
 functions enjoy the property that we can find the limit simply by susbstituting
 the 
\begin_inset Formula $x$
\end_inset

 value?
\begin_inset ERT
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\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
We can translate this into a mathematical expression:
\end_layout

\begin_layout Standard
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
begin{question}
\end_layout

\end_inset

Which functions 
\begin_inset Formula $\ec f$
\end_inset

 satisfy
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow a}f(x)=f(a)}?
\]

\end_inset


\begin_inset ERT
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\backslash
end{question}
\end_layout

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
We can turn this question into a definition.
\end_layout

\begin_layout BeginFrame
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{
\end_layout

\end_inset

Continuity at a point
\begin_inset ERT
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}
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\end_layout

\begin_layout Definition
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Continuity at a point)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

 A function 
\begin_inset Formula $\ec f$
\end_inset

 is 
\begin_inset ERT
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{
\end_layout

\end_inset


\begin_inset ERT
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\backslash
hilite
\end_layout

\end_inset

 continuous
\begin_inset ERT
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}
\end_layout

\end_inset

 at a point 
\begin_inset Formula $\ec x=a$
\end_inset

 if
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow a}f(x)=f(a)}.
\]

\end_inset

 That is, a function is 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

continuous
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

 at 
\begin_inset Formula $\ec x=a$
\end_inset

 if we can find the limit by evaluating 
\begin_inset Formula $\ec f(a)$
\end_inset

.
\end_layout

\begin_layout Standard
Geometrically, a function is continuous at 
\begin_inset Formula $\ec x=a$
\end_inset

 if it has no holes, jumps, or asymptotes at 
\begin_inset Formula $\ec x=a$
\end_inset

.
 
\end_layout

\begin_layout Itemize
If it has a hole at 
\begin_inset Formula $\ec x=a$
\end_inset

, then 
\begin_inset Formula $\ec f(a)$
\end_inset

 is undefined.
 
\end_layout

\begin_layout Itemize
If it has a jump at 
\begin_inset Formula $\ec x=a$
\end_inset

, then 
\begin_inset Formula $\ec f(a)$
\end_inset

 might be defined, but the left- and right-sided limits disagree, so 
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)$
\end_inset

 does not exist.
 
\end_layout

\begin_layout Itemize
If it has an asymptote at 
\begin_inset Formula $\ec x=a$
\end_inset

, then the limit is not finite.
 (We discuss this in a subsequent module.) 
\end_layout

\begin_layout Standard
\noindent
Either way, 
\begin_inset Formula $\ec\lim_{x\rightarrow a}f(x)\neq f(a)$
\end_inset

.
\end_layout

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\end_layout

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Some examples
\begin_inset ERT
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}
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\end_inset


\end_layout

\begin_layout Standard
As mentioned already, 
\begin_inset Formula $\ec f(x)=x^{2}$
\end_inset

 is continuous at 
\begin_inset Formula $\ec x=1$
\end_inset

 because
\begin_inset Formula 
\[
\ec{\lim_{x\rightarrow1}f(x)=f(1)}.
\]

\end_inset

 On the other hand, 
\begin_inset Formula $\ec s(x)=\sin\left(\frac{1}{x}\right)$
\end_inset

 is not continuous at 
\begin_inset Formula $\ec x=0$
\end_inset

 because
\begin_inset Formula 
\[
{\ec\lim_{x\rightarrow0}s(x)\neq s(0)};
\]

\end_inset

 in fact, 
\begin_inset Formula $\ec s(0)$
\end_inset

 does not even exist!
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
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\begin_layout Plain Layout


\backslash
begin{definition}
\end_layout

\end_inset

 If a function 
\begin_inset Formula $\ec f$
\end_inset

 is not continuous at 
\begin_inset Formula $\ec x=a$
\end_inset

 we say that 
\begin_inset Formula $\ec f$
\end_inset

 is 
\begin_inset ERT
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{
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\begin_inset ERT
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\end_layout

\end_inset

 discontinuous
\begin_inset ERT
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}
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\end_inset

 at 
\begin_inset Formula $\ec x=a$
\end_inset

.
 
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\backslash
end{definition}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
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{
\end_layout

\end_inset

Which functions are continuous on their domains?
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}
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\end_inset


\end_layout

\begin_layout Standard
You should remember the following fact.
 
\begin_inset ERT
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\backslash
begin{theorem}
\end_layout

\end_inset

 The following functions are continuous at every point in their domain:
 
\end_layout

\begin_layout Itemize
polynomials; 
\end_layout

\begin_layout Itemize
rational functions
\begin_inset Newline newline
\end_inset

(these have the form 
\begin_inset Formula $\ec p(x)/q(x)$
\end_inset

 where 
\begin_inset Formula $\ec p,q$
\end_inset

 are polynomials); 
\end_layout

\begin_layout Itemize
exponential functions (
\begin_inset Formula $\ec a^{x}$
\end_inset

 where 
\begin_inset Formula $\ec a>0$
\end_inset

 and 
\begin_inset Formula $\ec a\neq1$
\end_inset

); 
\end_layout

\begin_layout Itemize
logarithmic functions (
\begin_inset Formula $\ec\log_{a}x$
\end_inset

 where 
\begin_inset Formula $\ec a>0$
\end_inset

 and 
\begin_inset Formula $\ec a\neq1$
\end_inset

); 
\end_layout

\begin_layout Itemize
trigonometric functions; 
\end_layout

\begin_layout Itemize
sums, differences, and products of functions named above; 
\end_layout

\begin_layout Itemize
quotients 
\begin_inset Formula $\ec f(x)/g(x)$
\end_inset

 of functions named above,; and 
\end_layout

\begin_layout Itemize
composition 
\begin_inset Formula $\ec f(g(x))$
\end_inset

 of functions named above, 
\emph on
provided that the range of 
\begin_inset Formula $\ec g$
\end_inset

 is a subset of the domain of 
\begin_inset Formula $\ec f$
\end_inset


\emph default
.
 
\end_layout

\begin_layout Standard
\noindent
\begin_inset ERT
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\backslash
end{theorem}
\end_layout

\end_inset


\end_layout

\begin_layout BeginFrame
\begin_inset ERT
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{
\end_layout

\end_inset

Example
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
The following functions are continuous everywhere 
\emph on
on their domains
\emph default
.
 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec x^{3}-4x^{2}+3$
\end_inset

, because it is a polynomial; 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\left(x^{2}-x\right)/\left(x-1\right)$
\end_inset

, because it is a rational function; 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\sin(x)/(x^{2}+1)$
\end_inset

, because it is the quotient of a trigonometric function and a polynomial
 (the domain of the function is 
\begin_inset Formula $\ec\mathbb{R}$
\end_inset

); and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\sin(x)/(x^{2}-1)$
\end_inset

, because it is the quotient of a trigonometric function and a polynomial
 (the domain of the function is 
\begin_inset Formula $\ec\mathbb{R}\backslash\{-1,1\}$
\end_inset

); and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec\sin\left(x^{3}-4x^{2}+3\right)$
\end_inset

, because it is the composition of a trigonometric function and a polynomial,
 
\emph on
and
\emph default
 the range of 
\begin_inset Formula $\ec x^{3}-4x+3$
\end_inset

 is a subset of the domain of 
\begin_inset Formula $\ec\sin(x)$
\end_inset

.
 
\end_layout

\begin_layout BeginFrame
\noindent
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset

Isn't that everything?
\begin_inset ERT
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}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
Not all functions are continuous everywhere.
 An important function that you should know is the 
\begin_inset ERT
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{
\end_layout

\end_inset


\begin_inset ERT
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\backslash
hilite
\end_layout

\end_inset

 greatest integer function
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset

, also called the 
\begin_inset ERT
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\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
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\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 floor function
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

,
\begin_inset Formula 
\[
{\ec\lfloor x\rfloor=\mathrm{the\ largest\ integer\ less\ than\ or\ equal\ to\ }x}.
\]

\end_inset

 For example,
\begin_inset Formula 
\[
{\ec\lfloor3.9\rfloor=3}\qquad\mathrm{and}\qquad{\ec\lfloor-3.9\rfloor=-4}.
\]

\end_inset


\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\noindent
 (You evaluate 
\begin_inset Formula $\ec\lfloor a\rfloor$
\end_inset

 in SAGE for any value of 
\begin_inset Formula $\ec a$
\end_inset

 using the command 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 floor(a)
\family default
.)
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
Why is the floor function discontinuous? Estimate numerically 
\begin_inset Formula $\ec\lim_{x\rightarrow0^{+}}\lfloor x\rfloor$
\end_inset

 and 
\begin_inset Formula $\ec\lim_{x\rightarrow0^{-}}\lfloor x\rfloor$
\end_inset

.
 You will find that the limits from each side are different.
 This means that 
\emph on
the limit does not exist, even though the 
\begin_inset Formula $y$
\end_inset

 value exists!
\emph default
 That makes it impossible for 
\begin_inset Formula $\ec\lim_{x\rightarrow0}\lfloor x\rfloor=\lfloor0\rfloor$
\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
The domain of the floor function is 
\begin_inset Formula $\mathbb{R}$
\end_inset

, so the floor function is 
\emph on
not
\emph default
 continuous everywhere on its domain.
\end_layout

\begin_layout BeginFrame
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{
\end_layout

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SAGE and discontinuous functions
\begin_inset ERT
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}
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\end_inset


\end_layout

\begin_layout OverlayArea
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{
\backslash
textwidth}{6in}
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
Discontinuous functions are a weakness for SAGE (and for every computer
 algebra system).
 SAGE is not preprogrammed to look for discontinuities when you try to plot
 its graph.
\end_layout

\begin_layout Standard
\noindent
\begin_inset VSpace bigskip
\end_inset


\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<1>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
The following is the output of the command 
\family typewriter

\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
ttcolor
\end_layout

\end_inset

 plot(floor(x),-3,3,thickness=2)
\family default
:
\end_layout

\begin_layout Standard
\noindent
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
 
\end_layout

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/floorx.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\end_deeper
\begin_layout Pause

\end_layout

\begin_layout Only
\begin_inset ERT
status open

\begin_layout Plain Layout

<2>
\end_layout

\end_inset


\end_layout

\begin_deeper
\begin_layout Standard
\noindent
A more accurate graph would look like this:
\end_layout

\begin_layout Standard
\begin_inset Float figure
placement h!
wide false
sideways false
status open

\begin_layout Plain Layout
\align center
\begin_inset Graphics
	filename figures/floorx_good.png
	lyxscale 20
	scale 20

\end_inset

 
\end_layout

\end_inset


\end_layout

\begin_layout Standard
If a function is discontinuous, it is 
\emph on
your responsibility
\emph default
 to notice this, and to take the necessary precautions for obtaining a correct
 graph, limit, etc.
\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Continuous functions
\begin_inset ERT
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\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
If a function is continuous at 
\emph on
all
\emph default
 points in an interval 
\begin_inset Formula $\ec I$
\end_inset

, we say that the function is 
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout


\backslash
hilite
\end_layout

\end_inset

 continuous on 
\begin_inset Formula $I$
\end_inset


\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
\end_layout

\begin_layout Pause

\end_layout

\begin_layout Standard
\begin_inset VSpace bigskip
\end_inset

For example,
\end_layout

\begin_layout Itemize
polynomials are continuous on 
\begin_inset Formula $\ec\mathbb{R}=\left(-\infty,\infty\right)$
\end_inset

; 
\end_layout

\begin_layout Itemize
the function 
\begin_inset Formula $\ec1/x$
\end_inset

 is continuous on 
\begin_inset Formula $\ec\mathbb{R}\backslash\left\{ 0\right\} =\left(-\infty,0\right)\cup\left(0,\infty\right)$
\end_inset

; and 
\end_layout

\begin_layout Itemize
the function 
\begin_inset Formula $\ec\tan x$
\end_inset

 is continuous on 
\begin_inset Formula 
\[
{\ec\left\{ x\neq\frac{\pi}{2}+\pi k:\; k\in\mathbb{N}\right\} =\cdots\cup\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\cup\cdots}.
\]

\end_inset

 
\end_layout

\begin_layout Standard
\noindent
You should check that these are correct.
\end_layout

\begin_layout BeginFrame
Application: Intermediate Value Theorem
\end_layout

\begin_layout Standard
\noindent
An important application of continuity is the 
\begin_inset ERT
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{
\end_layout

\end_inset


\begin_inset ERT
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\backslash
hilite
\end_layout

\end_inset

 Intermediate Value Theorem
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset

.
\end_layout

\begin_layout Theorem
\begin_inset ERT
status open

\begin_layout Plain Layout

[
\end_layout

\end_inset

(Intermediate Value Theorem)
\begin_inset ERT
status open

\begin_layout Plain Layout

]
\end_layout

\end_inset

 If
\end_layout

\begin_deeper
\begin_layout Itemize
the function 
\begin_inset Formula $\ec f$
\end_inset

 is continuous on the interval 
\begin_inset Formula $\ec[a,b]$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f(a)\neq f(b)$
\end_inset

, 
\end_layout

\begin_layout Standard
then 
\end_layout

\begin_layout Itemize
for any 
\begin_inset Formula $y$
\end_inset

 value 
\begin_inset Formula $\ec C$
\end_inset

 between 
\begin_inset Formula $\ec f(a)$
\end_inset

 and 
\begin_inset Formula $\ec f(b)$
\end_inset

, 
\end_layout

\begin_layout Itemize
there exists a value 
\begin_inset Formula $\ec c$
\end_inset

 such that 
\end_layout

\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec a<c<b$
\end_inset

, and 
\end_layout

\begin_layout Itemize
\begin_inset Formula $\ec f(c)=C$
\end_inset

.
 
\end_layout

\end_deeper
\end_deeper
\begin_layout BeginFrame
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

{
\end_layout

\end_inset

Example of the Intermediate Value Theorem
\begin_inset ERT
status collapsed

\begin_layout Plain Layout

}
\end_layout

\end_inset


\end_layout

\begin_layout Standard
\noindent
To illustrate the Intermediate Value Theorem, let 
\begin_inset Formula $\ec f(x)=x^{2}$
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.
 
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Suppose 
\begin_inset Formula $\ec a=1$
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 and 
\begin_inset Formula $\ec b=2$
\end_inset

.
 
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\begin_layout Itemize
Since 
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\begin_deeper
\begin_layout Itemize
\begin_inset Formula $\ec f$
\end_inset

 is continuous, 
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\begin_layout Itemize
\begin_inset Formula $\ec f(a)=1\neq4=f(b)$
\end_inset

, 
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\begin_layout Itemize
and 
\begin_inset Formula $\ec\sqrt{2}$
\end_inset

 is between 
\begin_inset Formula $\ec1$
\end_inset

 and 
\begin_inset Formula $\ec4$
\end_inset

, 
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\end_deeper
\begin_layout Itemize
we can conclude that for 
\emph on
some
\emph default
 value 
\begin_inset Formula $\ec c\in\left(1,2\right)$
\end_inset

, 
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\begin_layout Itemize
\begin_inset Formula $\ec f(c)=\sqrt{2}$
\end_inset

.
 
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\begin_layout Standard
We will use this fact in another module to illustrate an application of
 the Intermediate Value Theorem.
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{
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Application: Intermediate Value Theorem
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\begin_layout Standard
Effectively, the Intermediate Value Theorem says that if 
\begin_inset Formula $\ec f$
\end_inset

 is continuous, then a particle that travels along the curve defined by
 
\begin_inset Formula $\ec f$
\end_inset

 does not have to 
\begin_inset Quotes eld
\end_inset

jump
\begin_inset Quotes erd
\end_inset

 to move between any two points of the curve.
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\noindent
\align center
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\emph on
This is not true for discontinuous functions!
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\begin_layout Standard
\noindent
Using the floor function, a particle has to 
\begin_inset Quotes eld
\end_inset

jump
\begin_inset Quotes erd
\end_inset

 to get from the point 
\begin_inset Formula $\ec(-0.1,-1)$
\end_inset

 to the point 
\begin_inset Formula $\ec(0,0)$
\end_inset

.
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\noindent
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placement h!
wide false
sideways false
status open

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\align center
\begin_inset Graphics
	filename figures/floorx_good.png
	lyxscale 20
	scale 20

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Conclusion
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}
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\begin_layout Itemize
A function is 
\emph on

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 continuous
\emph default
 if we can compute the two-sided limit simply by substituting the 
\begin_inset Formula $x$
\end_inset

 value into the function.
 
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\begin_layout Itemize
Not all functions are continuous everywhere on their domains.
 The 
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{
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 greatest-integer
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}
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, or 
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{
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 floor
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}
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\end_inset

 function, is an example of this.
 
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\begin_layout Itemize
An application of continuity is the 
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{
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 Intermediate Value Theorem
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.
 We will use this in a future module.
 
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\end_body
\end_document
