Can Derivatives Be Defined at Boundary Points?

[ToffeeC]

Something has been bugging as of late: usually, derivatives (ordinary and partial) are defined for interior points. However, I often come across statements in which they seem to also be defined for boundary points. For example, Leibniz' rule of integration, as usually stated, assumes some function f : [a,b] x [c,d] -> R has a continuous partial derivative (with respect to one of the variables) on its domain. But what does that mean for points which lie on the boundary of the square [a,b] x [c,d]? Does this simply mean that the partial derivative is continuous in the interior of [a,b] x [c,d] and has a limit at each boundary point? It'd be nice if someone could clarify this for me.

 

[micromass]

Hi ToffeeC!

I'm not 100% certain, but here's how I interpret it. You have a function f which has a continuous partial derivative. For this, you would assume that f is defined on a certain open domain D. So this defines a partial derivative $$\partial f/\partial x$$.

Then, we say that there is a $$[a,b]\times[c,d]\subseteq D$$ such that f and $$\partial f/\partial x$$ are continuous on this set. As you see, for the definition of $$\partial f/\partial x$$, we only needed the domain D, and we then state continuousness on a smaller rectangle.

 

[Stephen Tashi]

This is an excellent question, but we must distinguish between the statement that a function f has continuous partial derivatves on [a,b] x [c,d] and the statement that the domain of f is only [a,b]x[c,d]. Furthermore, an author could carelessly write: "Let f be a function with continuous partial derivatives that maps [a,b]x[c,d] to ..." and not mean to imply that the domain of f is only [a,b]x[c,d].

So if a respectable text seems to imply a derivative exists on a boundary point of the domain of a function, it would be worthwhile to check carefully if it that is actually what is being said.

 

[olivermsun]

I often come across statements in which they seem to also be defined for boundary points. For example, Leibniz' rule of integration, as usually stated, assumes some function f : [a,b] x [c,d] -> R has a continuous partial derivative (with respect to one of the variables) on its domain. But what does that mean for points which lie on the boundary of the square [a,b] x [c,d]?

Well if it states that the partial derivative exists even on the boundary, then is it a problem (as far as Leibniz' rule is concerned)?

 

[Stephen Tashi]

Well if it states that the partial derivative exists even on the boundary, then is it a problem (as far as Leibniz' rule is concerned)?

Since it is an incorrect statement (without defining some modified version of the definition of derivative), it would be a problem for any mathematics that might follow.

 

[olivermsun]

"A function f is defined and has continuous partial derivatives on some closed domain F."

Where does it say that f is not defined also on an open domain G containing the closed domain F or whatever else you would like it to have to have nice partial derivatives?

It just says you have the continuous partial derivatives in hand, which is what (I think anyway) you need to establish the Leibniz rule.

 

[Stephen Tashi]

It just says you have the continuous partial derivatives in hand, which is what (I think anyway) you need to establish the Leibniz rule.

I'm not sure which "it" you are referring to, but I think we all agree that well written mathematical materials do not state that a function has a partial derivative on a boundary point of its domain.

Returning to the original post, it is correct that the standard definition of partial derivative would preclude having a partial derivative on a boundary point of the domain of a function. That definition involves a limit and the epsilon-delta definition of limit has a clause that says "if |x - a| < delta then |f(x) - L| < epsilon. There is no exception granted to x's that are not in domain of the function.

 

[ToffeeC]

Hi all, thanks for your replies. I agree that taking the set [a,b] x [c,d] as contained in a larger open region is the most sensible interpretation. I have seen it implied, however, from what I would think are 'respectable sources' that this isn't necessarily the case. I wonder then if it's just carelessness or if they aren't limiting partial derivatives to be defined for interior points. After all, when talking about functions in C^1[a,b] for example, there is a derivative at the end points, which are boundary points, maybe it's the same sort of thing?