Math4121 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

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 $f'(x)=\lambda$.

QED

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

If

or

then

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 .

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

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

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

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,

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

so,

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

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

QED