One of the simple ways is finding the slope of all points and the linear slopes would be one of the most frequent ones because the difference between all points on a line will have the same value.

I have implemented this in a simple MATLAB example. I am sure that you can write a better one for your application.

I would propose that you can use the definitions to calculate the required quantities.

Say you have y=f(x),

Then its slope will be Slope S= dy/dx= y(n+1)- y(n)/delta x corresponding the y points.

Then you calculate the second derivative dS/ dx,

For the straight line parts the second derivative will be zero.

All the parts of the curves that have second derivative will be straight line portions.

Best wishes