Sum of functions bounded between 0 and 1? Let $y \in \mathbb{R}$ and $x \in \mathbb{R}$. I'm looking for two functions $f,g$ such that
$$
f(x)+g(y) \in [0,1]
$$
Do they exist? In positive case, do you have suggestions for what $f$ and $g$ could be? I'm not interested in $f,g$ taking a constant value or period functions.
The function $\tanh$ is neither constant nor periodic and takes values in the open interval $\ ]{-1},1[\ $. By choosing the constants $c_1$, $c_2$ properly you can then fabricate many pairs of functions
$$f_i(x):=c_i\bigl(1+\tanh(\alpha_i x+\beta_i)\bigr)\qquad(i=1,2)$$
that suit your needs.
What about $f(x)=g(x)=$$(-1)^x/2$?
$(tanh^2x/2)$ won't work because the range of $f+g$ is $[0,1)$, not $[0,1]$