Where can we find examples of smooth functions with compact support? Is there a book? Where can we find examples of smooth functions of compact support?
Is there a book? How to show that
$$
f(x)=\exp\left(-\frac{1}{x} \right) \exp\left(-\frac{1}{1-x}\right)
$$
is smooth?
 A: It should be $$ f(x) = \exp \left( \frac{-1}{x} \right)   \exp \left( \frac{1}{x-1} \right) , \; \; \; 0 < x < 1  $$ and
$f(x) = 0$ for all other $x$. Smoothness follows from smoothness for $\exp (-1/x), x > 0.$ 
Next, there is some constant $A = \int_0^1 f(x) dx.$ We get an almost step function by
$$  g(x) = \frac{1}{A} \int_{- \infty}^x f(t) dt $$
Then we get a plateau from
$$  h(x) = g(x) g(B - x)  $$ for some $B > 1.$
A: Here's an explanation for why the function you gave is smooth. Consider the function $f(x) = \exp(-1/x)$. 
Step 1: Show by induction that $f^{(n)}(x) = p_n(1/x) \exp(-1/x)$ for some polynomial $p_n(x)$.
Step 2: Show that $\lim_{x \to 0^+} p(1/x) e^{-1/x} = 0$ for any fixed polynomial $p(x)$.
Step 3: If we define a function $g(x) = 0$ for $x \leq 0$ and $g(x) = f(x)$ for $x>0$, then on $(0,\infty)$ and $(-\infty,0)$ the $n$th derivatives clearly exist and are continuous. You must show that the left and right derivatives of $g^{(n)}(x)$ definitionally are both zero at $x=0$, then, to conclude that $f(x)$ is smooth. Use step 2.
Step 4: Finally, the function $g(x)g(1-x)$ is equivalent to the function you asked about (I think you made a typo) and is smooth since it is made up of smooth functions. It has compact support by inspection.

For other examples of smooth, compactly support functions, consider the convolution of any compactly supported function with a smooth compactly supported function. That is, let $\phi(x) \in C_c^\infty(R)$ and let $f(x)$ and $\phi(x)$ have compact support in $B_r(x)$. Then 
$$g(x) = (f * \phi)(x) = \int_{-\infty}^{\infty} f(x-y)\phi(y) dy$$
is both smooth and compactly supported. (See the wikipedia article on mollifiers for similar examples.)
