Functions having derivative with value in $[0,1]$ and derivative function continuous and strictly monotone in $\mathbb{R}^M$.

Question

Can you draw some examples of functions from $\mathbb{R}$ to $\mathbb{R}$ such that

• the derivative at each point in the domain has value in $[0,1]$.

and

• the derivative function is continuous and strictly monotone in $\mathbb{R}$.

are all these functions unbounded above?

Answer 1

A classic one is the antiderivative of a sigmoid: $f : x \mapsto \log(1+e^x)$.

Note that all such functions are indeed unbounded above: as $f'$ is strictly increasing, there exists $x_0$ such that $f'(x_0) = a >0$, and then for all $x \ge x_0$, $f'(x) \ge a$. Thus for all $x \ge x_0$, $f(x) \ge f(x_0)+a(x-x_0)$, and this lower bound goes to infinity when $x \to \infty$.

Answer 2

Let $f(t)$ be a strictly monotone function with values in $[0, 1]$, such that integrals of the form $\int\limits_{-\infty}^xf(t)dt$ converge. Then, the function $g(x)$ defined by

$$g(x)=\int\limits_{-\infty}^xf(t)dt$$

satisfies your criteria.

As a specific example, take $f(t)=\frac{1}{1+exp(-t)}$. Then, $g(x)=\ln(e^x+1)$.

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Such a function $g$, indeed, has to be unbounded. Since $g'$ is strictly monotone, it has to reach some positive value somewhere: $g(x_0) = C > 0$, and (again by monotonicity) for $x > x_0$ we have $g'(x) > g'(x_0) = c$. Now, since (by Newton-Leibniz) $g(x) = g(x_0) + \int\limits_{x_0}^xg'(t)dt$, we can bound $g(x)$ from below:

$$g(x) = g(x_0) + \int\limits_{x_0}^xg'(t)dt = g(x_0) + \int\limits_{x_0}^xg'(t)dt > g(x_0) + \int\limits_{x_0}^xcdt > \\ > g(x_0) + c(x-x_0)$$

Since $g(x_0) + c(x-x_0)$ is clearly unbounded, $g$ is unbounded, too.