exercises:
(put notes on exercises from 3.7, 3.8
x3: show that f(x)={ 2x sin (1/x) - cos(1/x) (x \neq 0)
                   { 1/2 (x=0)
    is not the derivative of any function.
    (even though satisfies IVT: being a derivative
     is not as strong as being continuous,
     but it's still pretty rigid)
hint: consider F(x) = x^2 sin (1/x) & x \neq 0\\ 0 & x = 0
then F'(x) = f(x) for x \neq 0.
and F'(0)=0.
(lemma: if f,g continuous and f'(x) = g'(x) except maybe at one point,
 then f=g+C
 (more generally, if f,g continous and f' = g' almost everywhere, then
  f = g + C)
 pf: integrate.

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BTW, sin(sin x) has a really boring graph:
it just look like a flatten sin;
the dynamics are also pretty lame (well, interesting in a sense)
very -slow- convergence to 0, b/c $f'(0)=1$
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if $f(x)=y$ (so (x,y) on graph),
and...
...f'(x)>0
  then f''(x) and f^{-1}''(y) have opposite signs
...f'(x)<0
  then f''(x) and f^{-1}''(y) have the same signs

In particular,
    f is increasing and concave up
iff f^{-1} is decreasing and concave down

[make them state and prove analogous statements for other cases, namely:]
  f is decreasing and concave up
iff f^{-1} is decreasing and concave up
  f is decreasing and concave down
iff f^{-1} is decreasing and concave down
------------------------------------------
Here's an exercise in:
- understanding formal language
- dealing with functions as first-order objects

"Explain what is meant by the formula:
 $$|f|(x) := |f(x)| $$
 [It defines the ``absolute value, applied to a -function-, not just a
 number'']
Also, graph $|sin|$.
