Semi-differentiability

Let's say you have a bumpy road that you want to drive on. The road has lots of hills and valleys, so it's not very smooth. To drive on this road, your car needs to be able to handle all of the bumps and changes in the surface.

In math, we use something called a function to describe how things change. A function tells us how one thing changes when another thing changes. In this case, we can use a function to describe the bumps and hills on the road.

When we talk about semi-differentiability, we're really talking about how smooth the function is. A function is considered smooth or differentiable when it has no sudden changes or jumps, like a continuous line. But a function may have places where it's not smooth, like a sharp point, or a sudden drop, or a corner.

Semi-differentiability is like a compromise between a smooth function and a not-so-smooth one. In this case, we say that a function is semi-differentiable if it's smooth in some places, but not in others.

For example, if the bumps on the road are mainly small, then the function that describes them is mostly smooth. But if there are sudden dips or rises in the road, then the function becomes less smooth in those areas.

So, semi-differentiability is like having a road that is bumpy in some places, but still manageable by your car. It's not completely smooth, but it's also not too rough. In math terms, we say that a function is semi-differentiable if it has derivatives in some regions, but not in others.