Local inverse

Imagine you have a toy train track that goes in a big circle. You and your friend start at the same place on the track and go around in opposite directions. Eventually, you pass each other and continue on your way. Even though you started in the same place and went in opposite directions, you ended up in different places. That's because you went around the circle in different directions.

Now, let's think about this in math terms. Imagine you have a function that takes a number as input and gives you another number as output. Just like the toy train track, you can start at a certain input value and get an output value. But what if you want to know what input value you need to get a certain output value? That's where the inverse function comes in.

The inverse function does the opposite of the original function. It takes the output value as input and gives you the input value as output. It's like tracing your steps backwards along the train track to get back to where you started.

But sometimes, the original function doesn't have a well-behaved inverse function. That's where local inverse comes in. A local inverse is like a mini inverse function that only works in a small area around a certain input value. Just like how you and your friend ended up in different places on the toy train track even though you started in the same place, sometimes the function behaves differently in different areas. So you might need a local inverse function to figure out what input value you need to get a certain output value in a specific area.

In summary, a local inverse is a mini inverse function that helps you figure out what input value you need to get a certain output value in a specific area of a function.