Multiplicity and set of zeros. exercise:
Let us assume that the function f has derivatives of all orders.
Suppose that all zeros of $f$ have finite multiplicity. Let $a$ and $b$ be points of $A$, such that $a<b$ and neither point is a zero. Show that $f$ has at most finitely many zeros in $] a, b[$
(We say that a point $c$ is a root of $f(x)=0$ with multiplicity $m$, if $f^{(k)}(c)=0$ for $k=0, \ldots, m-1$ and $f^{(m)}(c) \neq 0 .$ As usual $f^{(0)}$ denotes $f$.)
lemma:
A zero of finite multiplicity is an isolated point of the set of zeros
proof:
Considering Taylor polynomial $E(h)=f(c+h)-\left(f(c)+\frac{1}{1 !} f^{\prime}(c) h+\frac{1}{2 !} f^{\prime \prime}(c) h^{2}+\cdots+\frac{1}{m !} f^{(m)}(c) h^{m}\right)$ and using L’Hopital’s Rule we can show that $\lim\limits_{h \rightarrow 0} \frac{E(h)}{h^{m}}=0$. Because of the multiplicity and for $h \neq 0$ we can write
$$
\frac{f(c+h)-f(c)}{h^{m}}=\frac{1}{m !} f^{(m)}(c)+\frac{E(h)}{h^{m}}
$$ and from this we can deduce the proof of the lemma.
So if we know if all zeros of $f$ have finite multiplicity then all zeros are isolated. Meanwhile we have bounded interval we can use Bolzano–Weierstrass theorem to show that if f have infinitely many zeros then at least one of the zeros should be limit point which contradicts to be isolated.
But why I do need $f(a)$, $f(b)$ not being zero which was declared in the exercise.
I am suspicious about using intermediate value theorem and taylor polynomial with remainder. But I don't know how to do it??
Remark: ( May be it can be some kind of hint which I cannot figure out: next exercise asks ::: In the previous exercise, if f (a) and f (b) have the same sign, show that the number of zeros in ]a, b[, counted by multiplicity, is even. If $f(a)$ and $f(b)$ have opposite signs, show that the number of zeros in ]a, b[, counted by multiplicity, is odd)

I think it's just to avoid a proliferation of cases in the proof. If you allow $f(a)=0$, for instance, you have to say "$n$ times right-differentiable" every time, instead of just "$n$ times differentiable". A function can have right-derivatives of all orders; for instance,$$f(x)=\begin{cases}
e^{-1/x^2}\sin\frac{1}{x}&\text{if }x\ne 0\\
     0&\text{if }x=0
\end{cases}$$
on $[0,1]$. This has an infinite number of zeroes, all of finite multiplicity except $f(0)$, which has infinite multiplicity. But then you have to define what the multiplicity of a zero is at the end-points of your interval. Not difficult, just messy.
