bjt - Why transistor is non-linear in Cutoff & Saturation mode?

I am imagining Linearity in the sense that

F(ax+by)=aF(x) + bF(y)

Now suppose I have BJT characteristic curve :

In Active region, The response is : Ic = F(Vce) = const. So it is linear.
In Cut off region, Response is : Ic = F(Vce) = 0.
- No matter how large the increase in Vce, Ic is still zero. It is Similar to Air gap. So it should be linear as well.
In Saturation region, Response is : Ic = const * Vce. With const being the slope of the curve. In reasonable Vce, I see this slope is of a straight line (not of a power function). So it is also linear in this region.

What was wrong in the above argument that I applied to BJT ?

Edit: Actually , function : y = b is non-linear map