Smooth function with specific features

Question

Im looking for a function that's continuous everywhere and whose second derivative is continuous everywhere with the following properties:

1. Near the origin, the function is approximately constant: $$f\approx c$$ with $c \in \mathbb{R}$
2. Far from the origin (outside some scale), the function is approximately linear: $$f(x) \approx m*x+b$$ with $m \neq 0$

So the function would look like

.

I just defined this function piecewise in mathematica as an example, but the second derivative is not continuous, because the curvy bits are quadratic and the flat bits are linear, so the second derivative is a series of steps.

I will eventually define a vector field from this scalar field which describes a flow in a flat manifold. Moreover, there is an energy functional that is defined in terms of the second derivatives of this scalar field, hence the reason for continuity.

I tried using bump functions and integrating backwards to get the scalar field I want, but the integrals became messy quite fast, as you might expect. Similarly, I tried using hyperbolic tangents, but I wasn't able to smoothly attach the pieces together.

I am also trying to minimize the magnitude of the second derivative (because it relates to the energy functional). So I am hoping to find a function that approaches the origin in a linear way, then curves in such a way that the second derivative has constant sign, then approaches zero again, then has constant opposite sign. Here is an example of such a second derivative using bump functions.

I tried integrating this and using the integration constants to satisfy my required properties, but the integrals are quite messy!

I appreciate any help!

Answer 1

I actually found a descent enough function. I started with the conditions on the second derivative I wanted to satisfy. So I construct the following function, which is just the sum of two gaussians, such that the entire function is odd about the origin $$ \alpha(x) = - e^{-\frac{(x+a)^2}{\sigma}} + e^{-\frac{(x-a)^2}{\sigma}}$$ with $a,\sigma \in \mathbb{R}$. Now, this function is what I want the second derivative to look like. To get the original function, I integrated this function twice to get

$$ \phi(x) = \frac{1}{2}\Bigg( e^{-\frac{(x-a)^2}{\sigma}} - e^{-\frac{(x+a)^2}{\sigma}}\Bigg)\sigma + C_1 + C_2 x + \frac{\sqrt{\pi \sigma}}{2}\Bigg( (a-x)ERF\bigg(\frac{a-x}{\sqrt{\sigma}}\bigg) - (a+x)ERF\bigg(\frac{a+x}{\sqrt{\sigma}}\bigg)\Bigg)$$

where $C_1, C_2$ are integration constants and $ERF$ is the error function from integrating a gaussian. Now, to have a linear function at large values of x and a flat function near the origin, we just massage the parameters until the function has vanishing derivative at the origin (flat) and constant non-zero at large x (linear). For example, I find $C_1=0$, $C_2 = 1.5$, and $a \approx 1.0087$ gives me:

Now, differentiating this twice yields the original function:

Hope this helps someone else in the future if you ever look for a function like this!

Answer 2

Is approximation of standard $f(x) = x \cdot (1 - e^{-x^2})$ good enough?

If not - then probably $x \cdot (1 - w(x)) + w(x)$ where $w(x)$ is bump function is the best we can get.