Why we can not differentiate functions that have closed ball domain?

Question

Why we can not differentiate functions that have closed ball domain?

I think this comes from the idea that we can not differentiate functions that have closed intervals domain, as in this case we can study differentiation on the open interval and then study differentiation from the left at the end point of the interval and study differentiation from right at the starting point of the interval. am I correct?

If so how can I explain this for functions having closed ball domain?

Thanks!

Answer 1

There's no intrinsic reason not to define a derivative of a function defined on a closed ball, at a point on the boundary on that ball.

In more detail, if $B = \{x\in\mathbb R^n: |x|\le R\}$ and $f:B\to \mathbb R^m$, then for any point $x_0$ domain of $f$ we can define that the derivative of $f$ at $x_0$ means a linear mapping $T$ such that $$ f(x) = f(x_0) + T(x-x_0) + o(x-x_0) $$ for $x\to x_0$ and $x\in B$.

Even at the boundary points of $B$, there are neighborhoods of the ball that are rich enough that the derivative is unique if it exists. (All that's needed for this is that every neighborhood of $x_0$ relative to $B$ contains points whose difference from $x_0$ span $\mathbb R^n$).

The reason why textbooks often don't do this is that most of what they want to say can be said quite fine while restricting our attention to open domains, and doing so is quite a bit simpler -- there are (literally!) fewer edge cases that need to be considered when stating the results.

For one example of one such corner case, consider the function $$ f(x,y) = \begin{cases} 0 & \text{if } y < x \\ 0 & \text{if } y > 4x \\ 0 & \text{if } x = y = 0 \\ 1 & \text{if } 2x < y < 3x \\ \text{undefined} & \text{otherwise} \end{cases} $$ whose partial derivatives exist and are continuous (indeed identically $0$) everywhere in the domain of $f$, but $f$ is not even continuous at $(0,0)$.

Answer 2

This really depends on what type of function and derivative you are thinking of. I assume you are thinking of the usual definition of the total derivative of a function between real vector spaces. In this case, your intuition is correct!

Consider some function f defined on some ball B(0,R) of radius R sitting inside Real n-space. Since any derivative( given it exists ) at a point p is a measure of change in the output of f coming from an infinitesimal change in the input around p, you have to be able to change the input approaching p from the direction you are measuring the change.

Suppose p is on the boundary of the ball. At p it makes sense to talk about directional derivatives in directions pointing into the ball, but not out of it. However since there are directions in which we cannot change the input ( or cannot approach p from ) we cannot express the rate of change at p in all directions, which is the data of the total derivative.

Another way of thinking about total derivatives is as a linearization( I.e. linear map )of the function, which approximates f around p, thus one needs to have a “tiny vector space” i.e an open ball, consisting of tangent vectors around p in order for f to have a chance to exist.

I hope that this comment was useful.