We will be focusing on sufficient conditions of differentiability of . The theorem says that if and exist and are continuous at point , then is differentiable at .
We have , which we know is partially differentiable with respect to and , but may not be differentiable in general.
Differentiability at point in the setting is described as
is the matrix of partial derivatives with respect to the independent variables in .
What does all this mean? This is something that confused me for some time, and is likely to be helpful for others with the same doubts.
This is the definition of differentiation. We say , if it exists, is the derivative of at . Let the limit be . We’re effectively saying that for any , if . Here . However, this is not what we SEEM to say through this definition. What we seem to be saying is for any and , there exists such that . Is there a difference? Yes. This will be illustrated below.
In our example of multi-variable differentiation, we’ve made a function of . Why is that? What we mean by that is not that can be any function of , like , where is a constant. What we mean is , although this is not obvious from the fact that is a function of . Why should we explicitly mention the fact that should tend to as ? Because in the original definition we have made the argument that for any and such that the difference between and is . Here, may not converge to ! We have just proven its existence, and none of its properties! For example, we could have said that for any and , . Here . We have proven the existence of .
It is only when we specify that that we make the definition of the derivative clear- that it is . Such a limit is defined when for any , there exists such that when .
Now we come back to sufficient conditions of differentiability. Let a function be differentiable with respect to and at . This implies
Adding these two, we get
Can we say
assuming and are small enough? We have
Now we use the property that the partial derivatives are continuous.
is the correction factor which is utimately rectified. Remember that here or . It can’t be a combination of both, as only partial derivatives are continuous.
We now have the formula
The rest of the proof is elementary, and can be found in any complex anaysis textbook (pg. 67 of Complex Variabes and Applications, Brown and Churchill). I have only explained the difficult step in the proof.
Reiterating the theorem, if is is partially differentiable with respect to its independent variables at a particular point, and all those partial derivatives are continuous, then is differentiable at that point.