Is there a continuous function with these properties (not piecewise) In short, I'm wondering if I can find a continuous non-piecewise function with these properties. I've found one that was close but not perfect. It's actually really useful to have something like this to scale it. Sorry about the lack of formatting (EDIT: Thank you)
$$\begin{aligned}
f(0) &= 1\\
f(x) &> 0\\
f'(0) &> 1\\
f'(x) &> 0\\
f''(0) &= 0\\
\lim_{x\to\infty} f '(x) &= 1 \\
\lim_{x\to -\infty} f(x) &= 0
\end{aligned}
$$
The second and fourth constraints were added afterwards, so keep that in mind while viewing comments/answers
Bonus points if you can find a function that has an elementary integral and that approaches y = x + 1
I've managed to get a function that displays all of the properties except the inflection at zero (and obviously the decrease in $f'(x)$), being hyperbolic (simply $\sqrt{1 + x^2 / 4}+ x / 2$).
The actual purpose isn't for mathematical purposes, it's for a program. In short I need to normalize a value so that it cannot drop below 0 but has no cap therein, and should approach additivity but with diminishing returns, rather than increasing that I would get with the hyperbolic one above. Thus, for most of the positive portion I need it to be concave down. If it's integrable I can do more things with it without breaking time-independence but most of the time it won't matter.

After playing around with various compositions and games, here's an analytic one:
$$f(x) = \frac 2 \pi \frac {e^x} {e^x + 1} x \arctan x + e^{2x - (2 + \frac 1 \pi)x^2}$$
The first part gives you the limits at $\pm \infty$ and the second part makes adjustments at $0$.

So what was this good for anyway? :)
