Suppose ddz(f(z)) exists at a point z. Is ddz(f(z)) continuous at z?
Suppose exists at a point z. Is continuous at z?
Answer & Explanation
2022-01-18Added 39 answers
Intuitively, you would maybe think so but the answer is no. As a counterexample, consider Then the derivative is Note that does not exist, even though .
2022-01-19Added 42 answers
If you are thinking to functions over the complex numbers, it is a celebrated result of Goursat that if the derivative exists over an open set, then the function is analytic over the open set; in particular its derivative is analytic as well, in particular differentiable and, therefore, continuous.
However, the derivative might not exist over an open set. An example is the function , which is only differentiable at zero; indeed, the Wirtinger derivative is
and a function can be differentiable (over the complex numbers) only where the Wirtinger derivative vanishes (it’s just the Cauchy-RIemann condition expressed in a different way). The derivative actually exists at zero, because
The derivative only exists at a point so it is continuous over its domain, but I guess it’s not the notion of continuity you were thinking to.
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