In my urban economics course, we looked at the preceding graph. My professor stated that a key assumption/reason the model is unstable at certain points is that the slopes differ (in absolute value) between two points on the same $y=a$ counter line (the red line in graph) (see the graphics below). Notice that the point on the left of the graph has a steep slope, while the point on the right has a lesser slope.
Is there a mathematical term for the case when the slope of a curve like this is equal at two different values of $x$ that yield the same $y$ value?
Thank you!
The everywhere differentiable functions $f:\Bbb R\to\Bbb R$ with $f(x)=f(y)\implies f'(x)=f'(y)$ are exactly the (not necessarily strictly) increasing/decreasing functions.
If you want $f(x)=f(y)\implies|f'(x)|=|f'(y)|$ then the characterization is not so easy anymore. All these functions look probably something like this:
They can have plateaus and the left and right slope are identical. I think there is no established name in the literature for such functions.
In your context I would say you are looking for even functions, i.e. if $a$ is the maximum, then there holds $f(a-x)=f(a+x)$.