Research diaries #8: Weakening differentiability

In high school we all learned about how to compute the derivative. Here we will look at what it means for a function to be differentiable and how to weaken it.

A derivative of a function at a point gives the slope of the tangent at that point. See the pretty figure below.

The blue line we can express as ax+b where a,b are two constants determined by making the line tangent to the black curve. The derivative is given by a

To make it more applied you can think of this graph as a function of position and time. The derivative at a give time then correspond to the speed at that time. Is a derivative always well-defined? No, Let's give an example:

The black curve is dented. It has a sharp point at the red spot. If we do the previous procedure to find the derivative the derivative could have 2 different values

We see that at the point we want to compute the derivative we could draw 2 different lines that are tangent in two different ways: one approaches from the left and the other approaches from the right. So the derivative is not well defined because the slope can have 2 different values. We call this function not smooth. However, these 2 lines still give us information about how steep the function is. This weaker type differentiability is called Lipschitz continuity. I will skip the mathematical formalities and you can keep your intuitive understanding.

So what do functions look like that are not Lipschitz continuous. A simple example is

A discontinuous curve

The disconintuity makes it so that I cannot draw this double cone shape as before.

In the domain of mathematical analysis there is a large field which considers weak versions of properties we considered in a basic calculus course. By weakening these properties we can somehow obtain better results because the conditions are not that severe. Specifically, in the field of differential equations this is a common approach to find solutions of complicated equations which only exist in some weak sense. Maybe that is a nice topic for a future post.

All images made by me ^^

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