MATH2340 Week 4

## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

| $x$  | \| | $2$  | $4$   | $6$   | $8$ | $10$ | $12$ |
            |-----|-|----|-----|-----|---|----|----|
            | $f(x)$ | \| | $13$ | $\pi$ | $\frac12$ | $1$ | $1$  | $3$  |
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x$ if $x$ is an integer, and $f(x) = t$ such that $t$ is the number of pixels used to render the letter $x$ in this font, if $x$ is not an integer.
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x^2 - \sin(x)$
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

$f:X\to Y$ has

*domain* $X$, the set of inputs,

*codomain* $Y$, the set that all the outputs belong to,

*range* (or *image*) $f(X)=\\{y\in Y$ such that $y=f(x)$ for some $x\in X\\}$.

Can use a table, a rule, a formula to define a function.

$f(x) = x^2 - \sin(x)$

Most of our functions will be defined by formulae.
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

A function always gives the same output for the same input.

If $x$ is fixed, then $f(x)$ is fixed.

$f$ is the function, $f(x)$ means "the function $f$ evaluated at the input $x$" means "the output of $f$ which corresponds to the input $x$".
            
            ---
            ## Functions
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

A function always gives the same output for the same input.

But different inputs could give the same output.

#### Example
            $f:\mathbb R\to\mathbb R$ defined by $f(x) = x^2$ has $f(-x)=f(x)$ for all $x$, so $f$ is not injective.

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.
            
            ---
            ## Functions: injectivity and surjectivity
            A function is a machine that takes in things and puts out things.
            Usually, the things are real numbers.

A function always gives the same output for the same input.

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.
            
            ---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.
            #### Examples
            Which of the following are injective, surjective, bijective on domain $\mathbb R$ and codomain $\mathbb R$?

$f(x) = x^3, \\;\\; g(x) = x^3-x, \\;\\; h(x) = \frac1{\lvert x \rvert +1}, \\;\\; p(x) = \sin(x)$,

$q(x) = 5, \\;\\; r(x) = x+\lceil x \rceil$.

(How) can you restrict their domains and codomains to make them bijective?
            
            ---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $f(x) = x^3$.

Every horizontal line crosses the curve $\leq1$ time, so $f$ is injective.

Every horizontal line crosses the curve $\geq1$ time, so $f$ is surjective.

$f$ is both injective and surjective, so it is bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x cubed](gfx/x3.png "Desmos plot of y=x^3")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $g(x) = x^3-x$.

Some horizontal lines cross the curve $>1$ time, so $g$ is not injective.

Every horizontal line crosses the curve $\geq1$ time, so $g$ is surjective.

$g$ is not injective, so it is not bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x cubed minus x](gfx/x3-x.png "Desmos plot of y=x^3-x")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $h:\mathbb R\to\mathbb R$ given by $h(x) = \frac1{\lvert x \rvert +1}$
            is neither injective nor surjective.
            But

$h:[0,\infty)\to(0,1]$ would be bijective.

$h:(-\infty,0]\to(0,1]$ would be bijective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=1 on (abs x plus 1)](gfx/1onabsxplus1.png "Desmos plot of y=1/(|x|+1)")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $p:\mathbb R\to\mathbb R$ given by $p(x) = \sin(x)$
            is neither injective nor surjective.
            But

$p:[\frac{-\pi}2,\frac\pi2]\to[-1,1]$ would be bijective.

$p:[\frac{-3\pi}2,\frac{-\pi}2]\to[-1,1]$ would be bijective.

$p:[-\pi,0]\to[-1,0]$ would be surjective but not injective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=sin(x)](gfx/sinx.png "Desmos plot of y=sin(x)")
            </div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $q:\mathbb R\to\mathbb R$ given by $q(x) = 5$
            is neither injective nor surjective.

To make $q$ bijective, we would have to its domain and codomain to single points.
            </div>
            <div style="flex:1;margin-left:2vw">

</div>
            </div>

---
            ## Functions: injectivity and surjectivity

#### Definition
            A function $f$ is *injective* if $f(x_1)=f(x_2) \implies x_1=x_2$.

A function $f:X\to Y$ is *surjective* if its range is its codomain; if $\\,\forall y \in Y,\\;\exists\\,x \in X$ such that $f(x)=y$.

A function is *bijective* if it is both injective and surjective.

#### Examples
            $r:\mathbb R\to\mathbb R$ given by $r(x) = x+\lceil x\rceil$
            is injective but not surjective.
            But

$r:(0,1]\to(1,2]$ would be bijective.

$r:(0,2]\to(1,2]\cup(3,4]$ would be bijective.

$r:(0,2]\to(1,4]$ would be injective but not surjective.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=x plus the ceiling function of x](gfx/xplusceilx.png "Desmos plot of y=x+ceil(x)")
            </div>
            </div>

---
            ## Limit estimation
            Let $f:\mathbb R \to\mathbb R$ and $a\in\mathbb R$.

Informally, $\displaystyle \ell=\lim_{x\to a}f(x)$ is "the value $f(x)$ gets close to when $x$ gets close to $a$".

We need a more formal definition of limits for this course, because we need to be able to *prove* what $\ell$ is.

We need to prove what $\ell$ is because estimating can result in mistakes.

[Excel file demonstrating the dangers of estimating limits](https://nc.dasmithmaths.com/index.php/s/H2HgtQMrWDxbiJB)
            
            ---
            ## Limits of real functions
            #### Definition
            Suppose $c\lt a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
            Then

$f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

<span style="color:var(--uswds-red-cool-20)">for all $\varepsilon>0$</span>,   <span style="color:var(--uswds-indigo-20)">$\exists \delta>0$ such that</span>   <span style="color:var(--uswds-green-cool-20)">$0 \lt \lvert x-a \rvert \lt \delta \implies$</span> <span style="color:var(--uswds-gold-20)">$\lvert f(x)-\ell \rvert < \varepsilon$</span>.

#### Or, in words,
            <span style="color:var(--uswds-red-cool-20)">No matter how small we pick $\varepsilon>0$</span>,

<span style="color:var(--uswds-indigo-20)">we can always go close enough to $a$ (ie. $\delta$ close) to ensure that</span>

#### Notation
            In this case, we write $\displaystyle \lim\_{x\to a} f(x) = \ell$.

[Excel file demonstrating how a limit is defined](https://nc.dasmithmaths.com/index.php/s/b29754YarxgnEsC)
            
            ---
            ## Limits of real functions
            #### Definition
            Suppose $c\lt a\lt d$ and $f:(c,a)\cup(a,d):\to\mathbb R$.
            Then

$f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Or, in words,
            <span style="color:var(--uswds-red-cool-20)">No matter how small we pick $\varepsilon>0$</span>,

<span style="color:var(--uswds-indigo-20)">we can always go close enough to $a$ (ie. $\delta$ close) to ensure that</span>

<span style="color:var(--uswds-green-cool-20)">all $x$ closer to $a$ (ie. within $\delta$ of $a$) give</span>
            <span style="color:var(--uswds-gold-20)">$f(x)$ within $\varepsilon$ of $\ell$</span>.
            #### How to prove convergence if you know the limit
            Construct a machine (a function) that <span style="color:var(--uswds-indigo-20)">tells you $\delta$</span> <span style="color:var(--uswds-red-cool-20)">for any given $\varepsilon$</span>, and that $\delta$ <span style="color:var(--uswds-gold-20)">d</span><span style="color:var(--uswds-green-cool-20)">o</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)">s</span><span style="color:var(--uswds-gold-20)"> t</span><span style="color:var(--uswds-green-cool-20)">h</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)"> j</span><span style="color:var(--uswds-gold-20)">o</span><span style="color:var(--uswds-green-cool-20)">b</span>.
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### How to prove convergence if you know the limit
            Construct a machine (a function) that <span style="color:var(--uswds-indigo-20)">tells you $\delta$</span> <span style="color:var(--uswds-red-cool-20)">for any given $\varepsilon$</span>, and that $\delta$ <span style="color:var(--uswds-gold-20)">d</span><span style="color:var(--uswds-green-cool-20)">o</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)">s</span><span style="color:var(--uswds-gold-20)"> t</span><span style="color:var(--uswds-green-cool-20)">h</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)"> j</span><span style="color:var(--uswds-gold-20)">o</span><span style="color:var(--uswds-green-cool-20)">b</span>.

#### Theorem
            If $f(x)=x^2$ then $\lim_{x\to0}f(x) = 0$.
            #### Proof
            For any $\varepsilon>0$, let $\delta=\sqrt\varepsilon$.
            If $\lvert x-0 \rvert \lt \delta$, then
            $\left\lvert f(x)-0 \right\rvert = \lvert x^2 \rvert = x^2 \lt \delta^2 = \varepsilon$.   □
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Theorem
            If $f(x)=x^2$ then $\lim_{x\to0}f(x) = 0$.
            #### Rough work
            I want to make
            $\left\lvert f(x)-0 \right\rvert \lt \varepsilon$.
            I calculate $\left\lvert f(x)-0 \right\rvert = \left\lvert x^2 \right\rvert = x^2$

and I'm allowed to insist $\lvert x-0 \rvert \lt \delta$, which implies $x^2 \lt \delta^2$.

So I just need $\delta^2 \leqslant \varepsilon$.
            So choose $\delta = \sqrt\varepsilon$.
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

and I'm allowed to insist $\lvert x-0 \rvert \lt \delta$, which implies $x^2 \lt \delta^2$.

So I just need $\delta^2 \leqslant \varepsilon$.
            So choose $\delta = \sqrt\varepsilon$.
            #### Proof
            For any $\varepsilon>0$, let $\delta=\sqrt\varepsilon$.
            If $\lvert x-0 \rvert \lt \delta$, then
            $\left\lvert f(x)-0 \right\rvert = \lvert x^2 \rvert = x^2 \lt \delta^2 = \varepsilon$.   □
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Theorem
            If $f(x)=\frac{x+1}{2-x}$ then $\lim_{x\to1}f(x) = 2$.
            #### Proof
            For any $\varepsilon>0$, let $\delta=\min\\{\frac\varepsilon6,\frac12\\}$.

If $\lvert x-1 \rvert \lt \delta$, then $6\lvert x-1 \rvert\lt\varepsilon$ and $\lvert 2-x \rvert = 2-x \gt \frac12$,

so
            $\left\lvert\frac{x+1}{2-x} - 2 \right\rvert = \frac{3\lvert x-1 \rvert}{\lvert 2-x \rvert} \lt \frac{3\lvert x-1 \rvert}{\frac12} = 6 \lvert x-1 \rvert\lt\varepsilon$.   □
            
            ---
            ## Limits of real functions
            #### Theorem
            If $f(x)=\frac{x+1}{2-x}$ then $\lim_{x\to1}f(x) = 2$.
            #### Rough work
            I want to make
            $\left\lvert f(x)-2 \right\rvert \lt \varepsilon$.
            I calculate $\left\lvert f(x)-2 \right\rvert = \ldots = \frac{3\lvert x-1 \rvert}{\lvert 2-x \rvert}$

and I'm allowed to insist $\lvert x-1 \rvert \lt \delta$.
            If $x$ is close to $1$ then $\lvert 2-x \rvert$ is close to $1$ too.

Quantitatively, if $\lvert x-1 \rvert \lt \frac12$, then $\frac12 \lt \lvert 2-x \rvert \lt \frac32$, so $\frac1{\lvert 2-x \rvert} \lt 2$.

That tells me $\left\lvert f(x)-2 \right\rvert = \frac{3\lvert x-1 \rvert}{\lvert 2-x \rvert} \lt 6\lvert x-1 \rvert$.

Now to achieve $\left\lvert f(x)-2 \right\rvert \lt \varepsilon$, all I need is $6\lvert x-1 \rvert \lt \varepsilon$; equivalently $\lvert x-1 \rvert \lt \frac\varepsilon6$.

To make all this work, I need both $\lvert x-1 \rvert \lt \frac12$ and $\lvert x-1 \rvert \lt \frac\varepsilon6$.

So choose $\delta = \min\\{\frac\varepsilon6,\frac12\\}$.
            
            ---
            ## Limits of real functions
            ### Why not input at the limit point?
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

for all $\varepsilon>0$,   $\exists \delta>0$ such that   $0 \lt \lvert x-a \rvert \lt \delta \implies$$\lvert f(x)-\ell \rvert < \varepsilon$.

Why $0 \lt \lvert x-a \rvert \lt \delta$ not just $\lvert x-a \rvert \lt \delta$?

Consider $\chi\_{\\{3\\}}:\mathbb R\to\mathbb R$, the indicator function of the set $\\{3\\}$, which is $0$ everywhere except $1$ if the input is $3$.
            </div>
            <div style="flex:1;margin-left:2vw">

![Desmos plot of y=indicator function for set {3} of x](gfx/chi3.png "Desmos plot of y=\chi_{3}(x)")
            </div>
            </div>

We want limits to be defined so that $\displaystyle \lim\_{x\to3}\chi\_{\\{3\\}}(x) = 0$.
            
            ---
            ## Limits of real functions
            #### Definition
            $f(x)$ *converges* to *limit* $\ell\in\mathbb R$ as $x\to a$ if

#### Theorem
            If $f(x)=5x+3-\chi\_{\\{2\\}}(x)$ then $f(2)=12$ but $\lim_{x\to2}f(x) = ?$.
            #### Proof
            $f(2) = 5(2)+3-\chi\_{\\{2\\}}(1) = 10+3-1=12$.

Your turn!
            $\ldots$
            #### How to prove convergence if you know the limit
            Construct a machine (a function) that <span style="color:var(--uswds-indigo-20)">tells you $\delta$</span> <span style="color:var(--uswds-red-cool-20)">for any given $\varepsilon$</span>, and that $\delta$ <span style="color:var(--uswds-gold-20)">d</span><span style="color:var(--uswds-green-cool-20)">o</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)">s</span><span style="color:var(--uswds-gold-20)"> t</span><span style="color:var(--uswds-green-cool-20)">h</span><span style="color:var(--uswds-gold-20)">e</span><span style="color:var(--uswds-green-cool-20)"> j</span><span style="color:var(--uswds-gold-20)">o</span><span style="color:var(--uswds-green-cool-20)">b</span>.
            
            ---
            ## Limits of real functions
            #### Theorem
            If $f(x)=5x+3-\chi\_{\\{2\\}}(x)$ then $f(2)=12$ but $\lim_{x\to2}f(x) = ?$.
            #### Rough work
            If $x\approx2$ but $x\neq2$, then $f(x)\approx 5 (2) + 3 - 0 = 13$, so guess that the limit is $13$.

I want to make
            $\left\lvert f(x)-13 \right\rvert \lt \varepsilon$.
            Suppose $x\neq2$.
            Then $f(x)=5x+3-0$.

I calculate $\left\lvert f(x)-13 \right\rvert = \left\lvert 5x+3-13 \right\rvert = 5\left\lvert x-2 \right\rvert$

and I'm allowed to insist $0 \lt \lvert x-2 \rvert \lt \delta$.
            Now to achieve $\left\lvert f(x)-13 \right\rvert \lt \varepsilon$, all I need is $5\lvert x-2 \rvert \lt \varepsilon$; equivalently $\lvert x-2 \rvert \lt \frac\varepsilon5$.
            So choose $\delta = \frac\varepsilon5$.
            
            ---
            ## Limits of real functions
            #### Theorem
            If $f(x)=5x+3-\chi\_{\\{2\\}}(x)$ then $f(2)=12$ but $\lim_{x\to2}f(x) = ?$.
            #### Rough work
            If $x\approx2$ but $x\neq2$, then $f(x)\approx 5 (2) + 3 - 0 = 13$, so guess that the limit is $13$.

I want to make
            $\left\lvert f(x)-13 \right\rvert \lt \varepsilon$.
            Suppose $x\neq2$.
            Then $f(x)=5x+3-0$.

I calculate $\left\lvert f(x)-13 \right\rvert = \left\lvert 5x+3-13 \right\rvert = 5\left\lvert x-2 \right\rvert$

and I'm allowed to insist $0 \lt \lvert x-2 \rvert \lt \delta$.
            Now to achieve $\left\lvert f(x)-13 \right\rvert \lt \varepsilon$, all I need is $5\lvert x-2 \rvert \lt \varepsilon$; equivalently $\lvert x-2 \rvert \lt \frac\varepsilon5$.
            So choose $\delta = \frac\varepsilon5$.
            #### Proof
            For any $\varepsilon>0$, let $\delta=\frac\varepsilon5$.
            If $0 \lt \lvert x-2 \rvert \lt \delta$, then $\left\lvert f(x)-13 \right\rvert = \left\lvert 5x+3-\chi\_{\\{2\\}}(x)-13 \right\rvert = 5 \lvert x-2 \rvert\lt5\frac\varepsilon5=\varepsilon$.   □
            
            ---
            ## Limits of real functions
            <div style="display:flex;flex-direction:row;width:100%">
            <div style="flex:1;margin-right:2vw;background-color:white;color:black">

Plot of $y=f(x)$.

![Desmos plot of y=x plus sgn of x minus 2](gfx/xplussgnxminus2.png "Desmos plot of y=x+sgn(x-2)")

</div>
            <div style="flex:2;margin-left:2vw">
              <div style="display:flex;flex-direction:column;height:100%">
                <div style="flex:1;">
                  <div style="display:flex;flex-direction:row;width:100%">
                    <div style="flex:1;margin-right:2vw">

$\displaystyle f(2)=$

$\displaystyle\lim\_{x\to2}f(x)=$

</div>
                    <div style="flex:1;margin-left:2vw">

Some content

</div>
                  </div>
                </div>
                <div style="flex:2;">

Some more content

</div>
              </div>
            </div>
            </div>