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Let $U$ be an open set in $\mathbb{C}$ and $f$ differentiable in the real sense. Prove that if $c$ is such that the following limit exists: $$\lim_{h→0}=\frac{|f(c+h)−f(c)|}{|h|}$$ Then $f$ is differentiable in the complex sense or $\bar{f}$ is differentiable in the complex sense.

$h$ is a complex number.

Ernesto Carro
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    $U$ never came into play. – Daniel Donnelly Sep 26 '13 at 04:13
  • Is it a hypothesis of more? – Ernesto Carro Sep 26 '13 at 12:07
  • I had thought to use matrix. We know that if f is differentiable in the real sense there exists $Df$, in particular exists $Df(c)$, my biggest question is: under the assumption of the limit, we can ensure that $|Df(c)(w)|$ = $|Df(c)(v)|$ ($v$ and $w$ unit vectors in $U$)? – Ernesto Carro Sep 26 '13 at 12:13
  • Possible duplicate of http://math.stackexchange.com/questions/29615/if-a-complex-function-f-is-real-differentiable-then-f-or-overlinef-are?rq=1 and http://math.stackexchange.com/questions/141294/existence-of-the-absolute-value-of-the-limit-implies-that-either-f-or-bar?rq=1 – Daniel R Sep 26 '13 at 18:38

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