# Lecture 4

## Chapter 5. Differentiation

### The continuity of the derivative

#### Theorem 5.12

Suppose $f$ is differentiable on $[a,b]$, Then $f'$ attains intermediate values between $f'(a)$ and $f'(b)$.

Proof:

Let $\lambda\in (f'(a),f'(b))$. We need to show that there exists $x\in (a,b)$ such that $f'(x)=\lambda$.

Let $g(x)=f(x)-\lambda x$. Then $g$ is differentiable on $(a,b)$ and

$$
g'(x)=f'(x)-\lambda.
$$

So $g'(a)=f'(a)-\lambda<0$ and $g'(b)=f'(b)-\lambda>0$.

We need to show that $g'(x)=0$ for some $x\in (a,b)$.

Since $g'(a)<0$, $\exists t_1\in (a,b)$ such that $g'(t_1)<g(a)$.

If not, then $g(t)\geq g(a)$ for all $t\in (a,b)$. But then $g'(a)\gets \frac{g(t)-g(a)}{t-a}\geq 0$, which contradicts $g'(a)<0$.

With the loss of generality, since $g'(b)>0$, $\exists t_2\in (a,b)$ such that $g'(t_2)<g(b)$.

Hence, $g$ attains its infimum on $[a,b]$ at some $x\in (a,b)$. Then this $x$ is a local minimum of $g$ on $(a,b)$.

So $g'(x)=0$ and $f'(x)=\lambda$.

EOP

### L'Hôpital's Rule

#### Theorem 5.13

Suppose $f$ and $g$ are differentiable on $(a,b)$ and $g'(x)\neq 0$ for all $x\in (a,b)$, where $-\infty\leq a<b\leq \infty$. Suppose

$$
\frac{f'(x)}{g'(x)}\to A \text{ as } x\to a\dots
$$

If

$$
f(x)\to 0 \text{ and } g(x)\to 0 \text{ as } x\to a,
$$

or

$$
g(x)\to \infty \text{ as } x\to a,
$$

then

$$
\frac{f(x)}{g(x)}\to A \text{ as } x\to a.
$$

Note that all these numbers $A$ can be $\infty$ or $-\infty$ (on extended real line).

We're using the open neighborhood definition of $\to$ here. An open neighborhood of $\infty$ is an interval of the form $(c,\infty)$ for some $c\in \mathbb{R}$.

> Recall the [Definition 3.1](https://notenextra.trance-0.com/Math4111/Math4111_L13#definition-31).

Proof:

Main step:

Suppose $-\infty\leq A\leq \infty$, and let $q>A$ with neighborhood $(-,\infty,q)$. Then $\exists c\in \mathbb{R}$ such that $\frac{f(x)}{g(x)}<q,\forall x\in (a,c)$.

Proof of the main step:

Fix $A<r<q$. Then $\exists c\in (a,b)$ such that $\frac{f'(x)}{g'(x)}<r,\forall x\in (a,c)$.

Now, for any $a<x<y<c$, by generalized mean value theorem, $\exists t\in (x,y)$ such that

$$
\frac{f(x)-f(y)}{g(x)-g(y)}=\frac{f'(t)}{g'(t)}
$$

Since $t\in (a,c)$, $\frac{f'(t)}{g'(t)}<r$.

Case 1: $f(x)\to 0$ and $g(x)\to 0$ as $x\to a$.

As $x\to a$, $f(x)\to 0$ and $g(x)\to 0$. So

$$
\begin{aligned}
\lim_{x\to a}\frac{f(x)-f(y)}{g(x)-g(y)}&=\lim_{x\to a}\frac{0-f(y)}{0-g(y)}\\
&=\lim_{x\to a}\frac{f(y)}{g(y)}\\
&=\frac{f'(y)}{g'(y)}\\
&\leq r<q
\end{aligned}
$$

$\forall y\in (a,c)$, $\frac{f(y)}{g(y)}<q$.

Case 2: $g(x)\to \infty$ as $x\to a$.

We can find $c_1\in (a,y)$ such that $g(x)>g(y)$ for all $x\in (a,c_1)$.

Therefore,

$$
\begin{aligned}
\frac{f(x)-f(y)}{g(x)}&<\frac{r[g(x)-g(y)]}{g(x)}\\
\frac{f(x)}{g(x)}&<r-\frac{rg(y)}{g(x)}+\frac{f(y)}{g(x)}
\end{aligned}
$$

To make the right side less than $q$, we need

$$
\frac{|rg(y)|+|f(y)|}{|g(x)|}<q-r
$$

so,

$$
|g(x)|>\frac{|rg(y)|+|f(y)|}{q-r}
$$

There exists $c_2\in (a,c_1)$ such that $|g(x)|>\frac{|rg(y)|+|f(y)|}{q-r},\forall x\in (a,c_2)$.

So $\forall x\in (a,c_2)$,

$$
\frac{f(x)}{g(x)}<\frac{rg(y)+f(y)}{g(x)}<r+(q-r)=q
$$

$\forall x\in (a,c_2)$, $\frac{f(x)}{g(x)}<q$.

EOP
