Hello,

I try to define a function $\displaystyle f :[a,a+1]-> [q,p] $, with $\displaystyle a \in \mathbb{Z}, q,p \in \mathbb{Q}$, s.t. f is smooth and has a critical value in $\displaystyle q$ and $\displaystyle p$.

Can you please help me? I try it with a lot functions, like this:

If we first assume that q< p:

$\displaystyle

f(t)=q+\frac{1}{exp(\frac{1}{(t-a)}-1)}*(p-q) $ for a+1>t>a and f(t)=q for t=a

but it doesn't work.

Ok this function is smooth and has a critical value at q. But not in p.

Thanks in advance!