Searching for a expression of a dampening function

Question

I am looking for a function $f(x)$ where

• $f(x)\rightarrow\infty$ for $x\rightarrow\infty$
• $f(x)=0$ for $x=0$
• $f'(x)=1$ for $x=0$, where $f'$ is the derivative of $f$ (about this I am not adamant)
• I don't care about $x<0$
• $f$ is strictly increasing for $x>0$
• $f$ is completely differentiable everywhere for $x>0$
• $f(x)<x$ for $x>0$

I would prefer a function closely related to a logarithmic function.

I want to use the function to make a positive parameter smaller, to give it sort of a constraint.

Answer 1

I got the idea, just after posting the question, but it might be useful for someone else later on, so I am posting it.

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The answer is, to simply use the shifted logarithm and divide it by the derivative of the original logarithm , so that the derivative becomes 1 for $x=0$:

$f(x) = log_b(x+1) \cdot ln(b) = ln(x+1)$

Answer 2

One of the simplest would be the square root function. If you want a logarithmic function, then ln(1+x) works. If you to get fancy, you could do the sigmoid function with the appropriate translation.