Can you give me an example of a real function with a behavior like this

Perhaps the cosine Fresnel integral $C(x)=\int_0^x\cos t^2\,dt$ is what you need:

There are elementary possibilities, too: $$2 \alpha \left(\frac{1}{e^{-\beta x}+1}-\frac{1}{2}\right) \left(1 +\frac{\sin (x)}{x}\right)$$

Plot with $\alpha=1,\beta=1$:

Plot with $\alpha=1/2,\beta=2$: