# Continuity

This section uses these add-on packages:

```julia
using CalculusWithJulia
using Plots
using SymPy
```

```julia; echo=false; results="hidden"
using CalculusWithJulia.WeaveSupport

const frontmatter = (
        title = "Continuity",
        description = "Calculus with Julia: Continuity",
        tags = ["CalculusWithJulia", "limits", "continuity"],
);

nothing
```

----

The definition Google finds for *continuous* is *forming an unbroken whole; without interruption*.

The concept in calculus, as transferred to functions, is
similar. Roughly speaking, a continuous function is one whose graph
could be drawn without having to lift (or interrupt) the pencil drawing it.

Consider these two graphs:

```julia; hold=true; echo=false
plt = plot([-1,0], [-1,-1],  color=:black, legend=false, linewidth=5)
plot!(plt, [0, 1], [ 1, 1], color=:black, linewidth=5)
plt
```

and

```julia; hold=true; echo=false
plot([-1,-.1, .1, 1], [-1,-1, 1, 1], color=:black, legend=false, linewidth=5)
```

Though similar at some level - they agree at nearly every value of
$x$ - the first has a "jump" from $-1$ to $1$ instead of the
transition in the second one. The first is not continuous at $0$ - a
break is needed to draw it - where as the second is continuous.



A formal definition of continuity was a bit harder to come about. At
[first](http://en.wikipedia.org/wiki/Intermediate_value_theorem) the
concept was that for any $y$ between any two values in the range for
$f(x)$, the function should take on the value $y$ for some $x$. Clearly
this could distinguish the two graphs above, as one takes no values in
$(-1,1)$, whereas the other - the continuous one - takes on all values in that range.


However, [Cauchy](http://en.wikipedia.org/wiki/Cours_d%27Analyse)
defined continuity by $f(x + \alpha) - f(x)$ being small whenever
$\alpha$ was small. This basically rules out "jumps" and proves more
useful as a tool to describe continuity.


The [modern](http://en.wikipedia.org/wiki/Continuous_function#History)
definition simply pushes the details to the definition of the limit:

> A function $f(x)$ is continuous at $x=c$ if $\lim_{x \rightarrow c}f(x) = f(c)$.

This says three things

* The limit exists at $c$.

* The function is defined at $c$ ($c$ is in the domain).

* The value of the limit is the same as $f(c)$.


This speaks to continuity at a point, we can extend this to continuity over an interval $(a,b)$ by saying:

> A function $f(x)$ is continuous over $(a,b)$ if at each point $c$ with $a < c < b$, $f(x)$ is continuous at $c$.

Finally, as with limits, it can be convenient to speak of *right*
continuity and *left* continuity at a point, where the limit in the
defintion is replaced by a right or left limit, as appropriate.


!!! warning
	The limit in the definition of continuity is the basic limit and not an extended sense where
	infinities are accounted for.

##### Examples of continuity

Most familiar functions are continuous everywhere.

* For example, a monomial function $f(x) = ax^n$ for non-negative, integer $n$ will be continuous. This is because the limit exists everywhere, the domain of $f$ is all $x$ and there are no jumps.

* Similarly, the basic trigonometric functions $\sin(x)$, $\cos(x)$ are continuous everywhere.

* So are the exponential functions $f(x) = a^x, a > 0$.

* The hyperbolic sine ($(e^x - e^{-x})/2$) and cosine ($(e^x + e^{-x})/2$) are, as $e^x$ is.

* The hyperbolic tangent is, as $\cosh(x) > 0$ for all $x$.

Some familiar functions are *mostly* continuous but not everywhere.

* For example, $f(x) = \sqrt{x}$ is continuous on $(0,\infty)$ and right continuous at $0$, but it is not defined for negative $x$, so can't possibly be continuous there.

* Similarly, $f(x) = \log(x)$ is continuous on $(0,\infty)$, but it is not defined at $x=0$, so is not right continuous at $0$.

* The tangent function $\tan(x) = \sin(x)/\cos(x)$ is continuous everywhere *except* the points $x$ with $\cos(x) = 0$ ($\pi/2 + k\pi, k$ an integer).

* The hyperbolic co-tangent is not continuous at $x=0$ -- when $\sinh$ is $0$,

* The semicircle $f(x) = \sqrt{1 - x^2}$ is *continuous* on $(-1, 1)$. It is not continuous at $-1$ and $1$, though it is right continuous at $-1$ and left continuous at $1$.

##### Examples of discontinuity

There are various reasons why a function may not be continuous.

* The function $f(x) = \sin(x)/x$ has a limit at $0$ but is not defined at $0$, so is not continuous at $0$. The function can be redefined to make it continuous.

* The function $f(x) = 1/x$ is continuous everywhere *except* $x=0$ where *no* limit exists.

* A rational function $f(x) = p(x)/q(x)$ will be continuous everywhere except where $q(x)=0$. (The function ``f`` may still have a limit where ``q`` is ``0``, should factors cancel, but ``f`` won't be defined at such values.)

* The function

```math
f(x) = \begin{cases}
         -1 & x < 0 \\
          0 & x = 0 \\
          1 & x > 0
\end{cases}
```

is implemented by `Julia`'s `sign` function. It has a value at $0$,
but no limit at $0$, so is not continuous at $0$. Furthermore, the
left and right limits exist at $0$ but are not equal to $f(0)$ so the
function is not left or right continuous at $0$. It is continuous everywhere except at $x=0$.

* Similarly, the function defined by this graph

```julia; hold=true; echo=false
plot([-1,-.01], [-1,-.01], legend=false, color=:black)
plot!([.01, 1], [.01, 1], color=:black)
scatter!([0], [1/2], markersize=5, markershape=:circle)
```

is not continuous at $x=0$. It has a limit of $0$ at $0$, a function
value $f(0) =1/2$, but the limit and the function value are not equal.

* The `floor` function, which rounds down to the nearest integer, is also not continuous at the integers, but is right continuous at the integers, as, for example, $\lim_{x \rightarrow 0+} f(x) = f(0)$. This graph emphasizes the right continuity by placing a point for the value of the function when there is a jump:

```julia; hold=true; echo=false
x = [0,1]; y=[0,0]
plt = plot(x.-2, y.-2, color=:black, legend=false)
plot!(plt, x.-1, y.-1, color=:black)
plot!(plt, x.-0, y.-0, color=:black)
plot!(plt, x.+1, y.+1, color=:black)
plot!(plt, x.+2, y.+2, color=:black)
scatter!(plt, [-2,-1,0,1,2], [-2,-1,0,1,2], markersize=5, markershape=:circle)
plt
```


* The function $f(x) = 1/x^2$ is not continuous at $x=0$: $f(x)$ is not defined at $x=0$ and $f(x)$ has no limit at $x=0$ (in the usual sense).

* On the Wikipedia page for [continuity](https://en.wikipedia.org/wiki/Continuous_function) the example of Dirichlet's function is given:

```math
f(x) =
\begin{cases}
0 & \text{if } x \text{ is irrational,}\\
1 & \text{if } x \text{ is rational.}
\end{cases}
```


The limit for any $c$ is discontinuous, as any interval about $c$ will
contain *both* rational and irrational numbers so the function will
not take values in a small neighborhood around any potential $L$.

##### Example

Let a function be defined by cases:

```math
f(x) = \begin{cases}
3x^2 + c & x \geq 0,\\
2x-3 & x < 0.
\end{cases}
```

What value of $c$ will make $f(x)$ a continuous function?

We note that for $x < 0$ and for $x > 0$ the function is a simple polynomial, so is continuous. At $x=0$ to be continuous we need a limit to exists and be equal to $f(0)$, which is $c$. A limit exists if the left and right limits are equal. This means we need to solve for $c$ to make the left and right limits equal. We do this next with a bit of overkill in this case:

```julia;
@syms x c
ex1 = 3x^2 + c
ex2 = 2x-3
del = limit(ex1, x=>0, dir="+") - limit(ex2, x=>0, dir="-")
```

We need to solve for $c$ to make `del` zero:

```julia;
solve(del, c)
```

This gives the value of $c$.

## Rules for continuity

As we've seen, functions can be combined in several ways. How do these relate with continuity?

Suppose $f(x)$ and $g(x)$ are both continuous on $I$. Then

* The function $h(x) = a f(x) + b g(x)$ is continuous on $I$ for any real numbers $a$ and $b$;

* The function $h(x) = f(x) \cdot g(x)$ is continuous on $I$; and

* The function $h(x) = f(x) / g(x)$ is continuous at all points $c$ in $I$ **where** $g(c) \neq 0$.

* The function $h(x) = f(g(x))$ is continuous at $x=c$ *if*  $g(x)$ is continuous at $c$ *and* $f(x)$ is continuous at $g(c)$.

So, continuity is preserved for all of the basic operations except when dividing by $0$.

##### Examples

* Since a monomial $f(x) = ax^n$ ($n$ a non-negative integer) is continuous, by the first rule, any polynomial will be continuous.

* Since both $f(x) = e^x$ and $g(x)=\sin(x)$ are continuous everywhere, so will be $h(x) = e^x \cdot \sin(x)$.

* Since $f(x) = e^x$ is continuous everywhere and $g(x) = -x$ is continuous everywhere, the composition $h(x) = e^{-x}$ will be continuous everywhere.

* Since $f(x) = x$ is continuous everywhere, the function $h(x) = 1/x$ - a ratio of continuous functions - will be continuous everywhere *except* possibly at $x=0$ (where it is not continuous).

* The function $h(x) = e^{x\log(x)}$ will be continuous on $(0,\infty)$, the same domain that $g(x) = x\log(x)$ is continuous. This function (also written as $x^x$) has a right limit at $0$ (of $1$), but is not right continuous, as $h(0)$ is not defined.


## Questions

###### Question

Let $f(x) = \sin(x)$ and $g(x) = \cos(x)$. Which of these is not continuous everywhere?

```math
f+g,~ f-g,~ f\cdot g,~ f\circ g,~ f/g
```

```julia; hold=true;  echo=false
choices = ["``f+g``", "``f-g``", "``f\\cdot g``", "``f\\circ g``", "``f/g``"]
answ = length(choices)
radioq(choices, answ)
```

###### Question

Let $f(x) = \sin(x)$, $g(x) = \sqrt{x}$.

When will $f\circ g$ be continuous?

```julia; hold=true;  echo=false
choices = [L"For all $x$", L"For all $x > 0$", L"For all $x$ where $\sin(x) > 0$"]
answ = 2
radioq(choices, answ, keep_order=true)
```

When will $g \circ f$ be continuous?

```julia; hold=true;  echo=false
choices = [L"For all $x$", L"For all $x > 0$", L"For all $x$ where $\sin(x) > 0$"]
answ = 3
radioq(choices, answ, keep_order=true)
```

###### Question

The composition $f\circ g$ will be continuous everywhere provided:

```julia; hold=true;  echo=false
choices = [
L"The function $g$ is continuous everywhere",
L"The function $f$ is continuous everywhere",
L"The function $g$ is continuous everywhere and $f$ is continuous on the range of $g$",
L"The function $f$ is continuous everywhere and $g$ is continuous on the range of $f$"]
answ = 3
radioq(choices, answ, keep_order=true)
```

######  Question

At which values is $f(x) = 1/\sqrt{x-2}$ not continuous?

```julia; hold=true;  echo=false
choices=[
L"When $x > 2$",
L"When $x \geq 2$",
L"When $x \leq 2$",
L"For $x \geq 0$"]
answ = 3
radioq(choices, answ)
```


###### Question

A value $x=c$ is a *removable singularity* for $f(x)$ if $f(x)$ is not
continuous at $c$ but will be if $f(c)$ is redefined to be $\lim_{x
\rightarrow c} f(x)$.


The function $f(x) = (x^2 - 4)/(x-2)$ has a removable singularity at
$x=2$. What value would we redefine $f(2)$ to be, to make $f$ a
continuous function?



```julia; hold=true;  echo=false
f(x) = (x^2 -4)/(x-2);
numericq(f(2.00001), .001)
```





###### Question

The highly oscillatory function

```math
f(x) = x^2 (\cos(1/x) - 1)
```

has a removable singularity at $x=0$. What value would we redefine
$f(0)$ to be, to make $f$ a continuous function?



```julia; hold=true;  echo=false
numericq(0, .001)
```


###### Question

Let $f(x)$ be defined by

```math
f(x) = \begin{cases}
c + \sin(2x - \pi/2) & x > 0\\
3x - 4 & x \leq 0.
\end{cases}
```

What value of $c$ will make $f(x)$ continuous?

```julia; hold=true;  echo=false
val = (3*0 - 4) - (sin(2*0 - pi/2))
numericq(val)
```

###### Question

Suppose $f(x)$, $g(x)$, and $h(x)$ are continuous functions on $(a,b)$. If $a < c < b$, are you sure that $lim_{x \rightarrow c} f(g(x))$ is $f(g(c))$?

```julia; hold=true;  echo=false
choices = [L"No, as $g(c)$ may not be in the interval $(a,b)$",
"Yes, composition of continuous functions results in a continuous function, so the limit is just the function value."
]
answ=1
radioq(choices, answ)
```

###### Question

Consider the function $f(x)$ given by the following graph

```julia; hold=true;  echo=false
xs = range(0, stop=2, length=50)
plot(xs, [sqrt(1 - (x-1)^2) for x in xs], legend=false, xlims=(0,4))
plot!([2,3], [1,0])
scatter!([3],[0], markersize=5)
plot!([3,4],[1,0])
scatter!([4],[0], markersize=5)
```

The function $f(x)$ is continuous at $x=1$?

```julia; hold=true;  echo=false
yesnoq(true)
```

The function $f(x)$ is continuous at $x=2$?

```julia; hold=true;  echo=false
yesnoq(false)
```

The function $f(x)$ is right continuous at $x=3$?

```julia; hold=true;  echo=false
yesnoq(false)
```

The function $f(x)$ is left continuous at $x=4$?

```julia; hold=true;  echo=false
yesnoq(true)
```

###### Question

Let $f(x)$ and $g(x)$ be continuous functions whose graph of $[0,1]$ is given by:

```julia; hold=true;  echo=false
xs = range(0, 1, length=251)
plot(xs, [sin.(2pi*xs) cos.(2pi*xs)], layout=2, title=["f" "g"], legend=false)
```

What is $\lim_{x \rightarrow 0.25} f(g(x))$?

```julia; hold=true;  echo=false
val = sin(2pi * cos(2pi * 1/4))
numericq(val)
```

What is $\lim{x \rightarrow 0.25} g(f(x))$?

```julia; hold=true;  echo=false
val = cos(2pi * sin(2pi * 1/4))
numericq(val)
```

What is $\lim_{x \rightarrow 0.5} f(g(x))$?

```julia; hold=true;  echo=false
choices = ["Can't tell",
"``-1.0``",
"``0.0``"
]
answ = 1
radioq(choices, answ)
```