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Math Help - Differentiable but not continuosly differentiable.

  1. #1
    Ant
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    Differentiable but not continuosly differentiable.

    Is it true that there are no function which are differentiable (in a complex sense) but not continuously differentiable (in a complex sense)? i.e. If a function f' exist then it is continuous.

    This seems to be the only correct conclusion from the theorems in my course but I've never heard it stated explicitly so I just wanted to check I'm not misunderstanding!

    Thanks,
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Differentiable but not continuosly differentiable.

    Yes.
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    Re: Differentiable but not continuosly differentiable.

    Being "differentiable" for functions of complex variables is much more severe than for real functions because of the increased "dimensionality". To be "differentiable" for functions of a real variable, we only have to have the limits "from the left" and "from the right" the same. To be "differentiable" for a function of a complex variable we must have all limits from the infinitely many ways to approach a point in two dimension the same. In fact, one can show that if a function of a complex variable is differentiable in some neighborhood of a point, then it is infinitely differentiable at that point.
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