What is a hump function? I have been in trouble with the hump function(s)
What are them?
Could you give me an explicit formula for "Hump"(not bump) function.
Thanks
 A: i think any function $g(x)$ that is the derivative of a sigmoid function
$$ g(x) = f'(x) $$
where $f(x)$ may be any of the functions shown as examples of an S-shaped sigmoid function indicated in the Wikipedia article, any of those can be legitimately called a "hump function".
A: If you're referring to the category of functions called humps that I'm familiar with, then of one these functions is $f$ defined below (try plotting it for $x \in [-2, 8]$, so that $f(x) \in [0, 25]$).
$$f(x) = \frac{1}{(x-3)^2+0.1} + \frac{1}{(x-2)^2+0.05} + 2$$
A: There are many a hump functions (so-called kernel functions): see the wikipedia.
The functions in examples above are not compact, i.e. for all $x\in(-\infty, \infty),\;\, f(x) \not= 0.$
If you want a compact function, where non-zero value is defined only in a line segment:
$$
f(x)= 
\begin{cases}
\frac{(\Delta^2 - x^2)^3}{\Delta},\quad |x|<\Delta\\
0, \quad\quad\quad\;\;\,|x|\geq \Delta.
\end{cases}
$$
This function has continuous derivatives. Therefore, it $f(x)$ is a smooth function.
