Semi-differentiability

In calculus, we learn how functions change and how to find their slopes or rates of change. One key idea is differentiability, which means we can find a derivative at every point of a function. But sometimes, a function might not have a derivative everywhere, so we need simpler ideas to still understand its behavior.

The ideas of one-sided differentiability and semi-differentiability help us understand a function’s slope when we only look at one side of a point. For a real-valued function f of a real variable, we say it is right differentiable at a point a if a derivative exists as the function’s input x approaches a from the right. Similarly, it is left differentiable at a if the derivative exists as x approaches a from the left.

These ideas are simpler than full differentiability. They are useful in many areas of mathematics where functions might not be smooth everywhere, but we can still learn important information by looking at their behavior from just one side. Understanding semi-differentiability helps mathematicians and scientists solve more complex problems.

One-dimensional case

In mathematics, a left derivative and a right derivative are special types of derivatives. They show how a function changes when we move just to the left or just to the right of a point.

A function is right differentiable at a point if we can find a derivative as we move toward that point from the right. It is left differentiable if we can find a derivative moving toward the point from the left. When both left and right derivatives exist at a point, the function is called semi-differentiable there. An example of this is the absolute value function at zero.

Properties

Any convex function on a convex open subset of Rⁿ is semi-differentiable. While every semi-differentiable function with one variable is continuous, this is not always true for functions with more variables.

Generalization

We can study functions that do more than just give out simple numbers. Some functions can give out lists of numbers or points in space. This helps mathematicians understand how these more complex functions change and act.