Semi-differentiability

In calculus, we study how functions change and how we can understand their slopes or rates of change. One important 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, and we need weaker ideas to still learn about its behavior.

The notions 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 weaker than full differentiability, meaning they require less for a derivative to exist. 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 analyze more complex problems.

One-dimensional case

In mathematics, a left derivative and a right derivative are special types of derivatives that look at how a function changes when we move just to the left or just to the right of a point. These ideas help us understand the behavior of functions in one direction only.

A function is right differentiable at a point if we can find a derivative as we move toward that point from the right. Similarly, 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, where the left derivative is -1 and the right derivative is 1.

Properties

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

Generalization

Instead of looking at functions that give out simple numbers, we can think about functions that give out more complex things, like lists of numbers or points in space. This idea helps mathematicians study how these more complicated functions change and behave.